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# 12-3. Continuity

Continuity
In this section, you will:
• Determine whether a function is continuous at a number.
• Determine the numbers for which a function is discontinuous.
• Determine whether a function is continuous.

Arizona is known for its dry heat. On a particular day, the temperature might rise as high as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 118 ∘ F  and drop down only to a brisk<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 95 ∘ F.  [link] shows the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>T</mi><mo>,</mo></annotation-xml></semantics>[/itex] where the output of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] T( x )  is the temperature in Fahrenheit degrees and the input<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is the time of day, using a 24-hour clock on a particular summer day.

<figure class="small" id="CNX_Precalc_Figure_12_03_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>Temperature as a function of time forms a continuous function.</figcaption> </figure>

When we analyze this graph, we notice a specific characteristic. There are no breaks in the graph. We could trace the graph without picking up our pencil. This single observation tells us a great deal about the function. In this section, we will investigate functions with and without breaks.

# Determining Whether a Function Is Continuous at a Number

Let’s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in [link] indicates that, at 2 a.m., the temperature was<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 96 ∘ F. By 2 p.m. the temperature had risen to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 116 ∘ F,  and by 4 p.m. it was<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 118 ∘ F.  Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 110.5 ∘ F.  In fact, any temperature between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 96 ∘ F  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 118 ∘ F  occurred at some point that day. This means all real numbers in the output between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 96 ∘ F  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 118 ∘ F  are generated at some point by the function according to the intermediate value theorem,

Look again at [link]. There are no breaks in the function’s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a continuous function. Temperature as a function of time is an example of a continuous function.

If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let’s create the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>D</mi><mo>,</mo></annotation-xml></semantics>[/itex] where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] D( x )  is the output representing cost in dollars for parking<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]number of hours. See [link].

Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a$25 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is$16. We can never be charged $13,$14, or \$15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function’s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.

<figure id="CNX_Precalc_Figure_12_03_002"> <figcaption>Parking-garage charges form a discontinuous function.</figcaption> </figure>

A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example.

A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function.

So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let’s use the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] y=f( x )  represented in [link] as an example.

<figure class="small" id="CNX_Precalc_Figure_12_03_003"></figure>

Condition 1 According to Condition 1, the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( a )  defined at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  must exist. In other words, there is a y-coordinate at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  as in [link].

<figure class="small" id="CNX_Precalc_Figure_12_03_004"></figure>

Condition 2 According to Condition 2, at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  the limit, written<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x) ,must exist. This means that at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  the left-hand limit must equal the right-hand limit. Notice as the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]in [link] approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  from the left and right, the samey-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

Condition 3 According to Condition 3, the corresponding<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>y</mi><mtext> </mtext></annotation-xml></semantics>[/itex]coordinate at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  fills in the hole in the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f.  This is written<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)=f(a).

Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in [link] so the function is continuous as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

<figure class="small" id="CNX_Precalc_Figure_12_03_005"> <figcaption>All three conditions are satisfied. The function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.</figcaption> </figure>

[link] through [link] provide several examples of graphs of functions that are not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  and the condition or conditions that fail.

<figure class="small" id="CNX_Precalc_Figure_12_03_006"> <figcaption>Condition 2 is satisfied. Conditions 1 and 3 both fail.</figcaption> </figure> <figure class="small" id="CNX_Precalc_Figure_12_03_007"> <figcaption>Conditions 1 and 2 are both satisfied. Condition 3 fails.</figcaption> </figure> <figure class="small" id="CNX_Precalc_Figure_12_03_008"> <figcaption>Condition 1 is satisfied. Conditions 2 and 3 fail.</figcaption> </figure> <figure class="small" id="CNX_Precalc_Figure_12_03_009"> <figcaption>Conditions 1, 2, and 3 all fail.</figcaption> </figure>
Definition of Continuity

A function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  provided all three of the following conditions hold true:

Condition 1:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(a)  exists.
Condition 2:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)  exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.
Condition 3:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)=f(a).

If a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a ,the function is discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

## Identifying a Jump Discontinuity

Discontinuity can occur in different ways. We saw in the previous section that a function could have a left-hand limit and aright-hand limit even if they are not equal. If the left- and right-hand limits exist but are different, the graph “jumps” at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a. The function is said to have a jump discontinuity.

As an example, look at the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] y=f( x )  in [link]. Notice as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics>[/itex]how the output approaches different values from the left and from the right.

<figure class="small" id="CNX_Precalc_Figure_12_03_010"> <figcaption>Graph of a function with a jump discontinuity.</figcaption> </figure>
Jump Discontinuity

A function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  has a jump discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  if the left- and right-hand limits both exist but are not equal:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ a − f(x)≠ lim x→ a + f(x).

## Identifying Removable Discontinuity

Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let’s look at the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] y=f( x )  represented by the graph in [link]. The function has a limit. However, there is a hole at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a. The hole can be filled by extending the domain to include the input<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  and defining the corresponding output of the function at that value as the limit of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

<figure class="small" id="CNX_Precalc_Figure_12_03_011"> <figcaption>Graph of function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]with a removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.</figcaption> </figure>
Removable Discontinuity

A function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  has a removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  if the limit,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x) , exists, but either

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] a )  does not exist or
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] a ), the value of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  does not equal the limit,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(a)≠ lim x→a f(x).

Identifying Discontinuities

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] x 2 −2x−15 x−5
2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics>[/itex] x+1, x<2 −x, x≥2
1. Notice that the function is defined everywhere except at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5.

Thus,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( 5 )  does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches 5 is 8, and Condition 2 is not satisfied.This means there is a removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5.

2. Condition 2 is satisfied because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] g(2)=−2.

Notice that the function is a piecewise function, and for each piece, the function is defined everywhere on its domain. Let’s examine Condition 1 by determining the left- and right-hand limits as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches 2.

Left-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 2 − ( x+1 )=2+1=3.  The left-hand limit exists.

Right-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 2 + ( −x )=−2.  The right-hand limit exists. But

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics>[/itex] x→ 2 − f(x)≠ lim x→ 2 + f(x).

So,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→2 f(x)  does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2.

Identify all discontinuities for the following functions as either a jump or a removable discontinuity.

1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] x 2 −6x x−6
2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics>[/itex] x , 0≤x<4 2x, x≥4
1. removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=6;
2. jump discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4

# Recognizing Continuous and Discontinuous Real-Number Functions

Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is the same as the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.  So<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)=f(a).  There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.

Examples of Continuous Functions

The following functions are continuous everywhere:

 Polynomial functions Ex:[/itex] f(x)= x 4 −9 x 2 Exponential functions Ex:[/itex] f(x)= 4 x+2 −5 Sine functions Ex:[/itex] f(x)=sin( 2x )−4 Cosine functions Ex:[/itex] f(x)=−cos( x+ π 3 )

The following functions are continuous everywhere they are defined on their domain:

 Logarithmic functions Ex:[/itex] f(x)=2ln( x ),[/itex] x>0 Tangent functions Ex:[/itex] f(x)=tan( x )+2,[/itex] x≠ π 2 +kπ,k[/itex]is an integer Rational functions Ex:[/itex] f(x)= x 2 −25 x−7 ,[/itex] x≠7

Given a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x ), determine if the function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

1. Check Condition 1:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(a)  exists.
2. Check Condition 2:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)  exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.
3. Check Condition 3:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)=f(a).
4. If all three conditions are satisfied, the function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.  If any one of the conditions is not satisfied, the function is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.
Determining Whether a Piecewise Function is Continuous at a Given Number

Determine whether the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)={ 4x, x≤3 8+x, x>3  is continuous at

1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow></annotation-xml></semantics>[/itex]
2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 8 3

To determine if the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a,  we will determine if the three conditions of continuity are satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

1. Condition 1: Does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(a)  exist?

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>12</mn></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] ⇒Condition 1 is satisfied.

Condition 2: Does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→3 f(x)  exist?

To the left of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=4x;  to the right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=8+x.  We need to evaluate the left- and right-hand limits as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches 1.

Left-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 3 − f(x)= lim x→ 3 − 4(3)=12
Right-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 3 + f(x)= lim x→ 3 + ( 8+x )=8+3=11

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 1 − f(x)≠ lim x→ 1 + f(x),<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→1 f(x)  does not exist.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">⇒</mo><mtext> Condition 2 fails</mtext><mtext>.</mtext></mrow></annotation-xml></semantics>[/itex]

There is no need to proceed further. Condition 2 fails at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3.  If any of the conditions of continuity are not satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3,  the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3.

2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 8 3

Condition 1: Does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( 8 3 )  exist?

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] 8 3 )=4( 8 3 )= 32 3 ⇒Condition 1 is satisfied.

Condition 2: Does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 8 3 f(x)  exist?

To the left of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x= 8 3 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=4x;  to the right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x= 8 3 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=8+x.  We need to evaluate the left- and right-hand limits as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 8 3 .

Left-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 8 3 − f(x)= lim x→ 8 3 − 4( 8 3 )= 32 3
Right-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 8 3 + f(x)= lim x→ 8 3 + ( 8+x )=8+ 8 3 = 32 3

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 8 3 f(x)  exists,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">⇒</mo><mtext>Condition 2 is satisfied</mtext><mo>.</mo></mrow></annotation-xml></semantics>[/itex]

Condition 3: Is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( 8 3 )= lim x→ 8 3 f(x)?

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] 32 3 )= 32 3 = lim x→ 8 3 f(x) ⇒Condition 3 is satisfied.

Because all three conditions of continuity are satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x= 8 3 ,  the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x= 8 3 .

Determine whether the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)={ 1 x , x≤2 9x−11.5, x>2  is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2.

yes

Determining Whether a Rational Function is Continuous at a Given Number

Determine whether the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)= x 2 −25 x−5  is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5.

To determine if the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5, we will determine if the three conditions of continuity are satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5.

Condition 1:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mtext> does not exist</mtext><mtext>.</mtext></mtd></mtr></mtable></annotation-xml></semantics>[/itex] ⇒Condition 1 fails.

There is no need to proceed further. Condition 2 fails at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5.  If any of the conditions of continuity are not satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5,the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5.

Analysis

See [link]. Notice that for Condition 2 we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] x→5 x 2 −25 x−5 = lim x→3 (x−5) (x+5) x−5                     = lim x→5 (x+5)                     =5+5=10                    ⇒Condition 2 is satisfied.

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=5,there exists a removable discontinuity. See [link].

<figure class="small" id="CNX_Precalc_Figure_12_03_013"></figure>

Determine whether the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)= 9− x 2 x 2 −3x  is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3.  If not, state the type of discontinuity.

No, the function is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3.  There exists a removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=3.

# Determining the Input Values for Which a Function Is Discontinuous

Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up.

To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.

Given a piecewise function, determine whether it is continuous at the boundary points.

For each boundary point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics>[/itex]of the piecewise function, determine the left- and right-hand limits as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] a,as well as the function value at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] a.
Check each condition for each value to determine if all three conditions are satisfied.
Determine whether each value satisfies condition 1:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(a)  exists.
Determine whether each value satisfies condition 2:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)  exists.
Determine whether each value satisfies condition 3:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)=f(a).
If all three conditions are satisfied, the function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.  If any one of the conditions fails, the function is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.

Determining the Input Values for Which a Piecewise Function Is Discontinuous

Determine whether the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is discontinuous for any real numbers.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mi>x</mi><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics>[/itex] x+1, x<2 3, 2≤x<4 x 2 −11, x≥4

The piecewise function is defined by three functions, which are all polynomial functions,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=x+1  on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x<2,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=3  on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 2≤x<4,and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)= x 2 −5  on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x≥4.  Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4.

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2, let us check the three conditions of continuity.

Condition 1:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics>[/itex] 2 )=3 ⇒Condition 1 is satisfied.

Condition 2: Because a different function defines the output left and right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2,does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 2 − f(x)= lim x→ 2 + f(x)?

Left-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 2 − f(x)= lim x→ 2 − ( x+1 )=2+1=3
Right-hand limit: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 2 + f(x)= lim x→ 2 + 3=3

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 3=3,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 2 − f(x)= lim x→ 2 + f(x)

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">⇒</mo><mtext>Condition 2 is satisfied</mtext><mo>.</mo></mrow></annotation-xml></semantics>[/itex]

Condition 3:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><munder><mrow><mi>lim</mi></mrow></munder></mtd></mtr></mtable></annotation-xml></semantics>[/itex] x→2 f(x)=3=f(2) ⇒Condition 3 is satisfied.

Because all three conditions are satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2,the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2.

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4,let us check the three conditions of continuity.

Condition 2: Because a different function defines the output left and right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4,does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 4 − f(x)= lim x→ 4 + f(x)?

Left-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 4 − f(x)= lim x→ 4 − 3=3
Right-hand limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 4 + f(x)= lim x→ 4 + ( x 2 −11 )= 4 2 −11=5

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] 3≠5,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ 4 − f(x)≠ lim x→ 4 + f(x) ,so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→4 f(x)  does not exist.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">⇒</mo><mtext>Condition 2 fails</mtext><mo>.</mo></mrow></annotation-xml></semantics>[/itex]

Because one of the three conditions does not hold at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4,the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)  is discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4.

Analysis

See [link]. At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4,there exists a jump discontinuity. Notice that the function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2.

<figure class="small" id="CNX_Precalc_Figure_12_03_014"> <figcaption>Graph is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2  but shows a jump discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=4.</figcaption> </figure>

Determine where the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)={ πx 4 ,  x<2 π x ,    2≤x≤6 2πx,  x>6  is discontinuous.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>6</mn></mrow></annotation-xml></semantics>[/itex]

# Determining Whether a Function Is Continuous

To determine whether a piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.

Given a piecewise function, determine whether it is continuous.

1. Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?
2. For each boundary point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  of the piecewise function, determine if each of the three conditions hold.
Determining Whether a Piecewise Function Is Continuous

Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mi>x</mi><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics>[/itex] sin(x), x<0 x 3 , x>0

The two functions composing this piecewise function are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=sin(x)  on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x<0  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)= x 3  on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x>0.  The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0,let us check the three conditions of continuity.

Condition 1:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> does not exist</mtext><mo>.</mo></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics>[/itex] ⇒Condition 1 fails.

Because all three conditions are not satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0, the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)  is discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0.

Analysis

See [link]. There exists a removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→0 f(x)=0, thus the limit exists and is finite, but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( a )  does not exist.

<figure class="small" id="CNX_Precalc_Figure_12_03_015"> <figcaption>Function has removable discontinuity at 0.</figcaption> </figure>

Access these online resources for additional instruction and practice with continuity.

# Key Concepts

• A continuous function can be represented by a graph without holes or breaks.
• A function whose graph has holes is a discontinuous function.
• A function is continuous at a particular number if three conditions are met:
• Condition 1:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(a)  exists.
• Condition 2:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)  exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a.
• Condition 3:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→a f(x)=f(a).
• A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
• A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See[link].
• Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See [link] and [link].
• For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See [link] and [link].

# Section Exercises

## Verbal

State in your own words what it means for a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]to be continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=c.

Informally, if a function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=c , then there is no break in the graph of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( c ), and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( c )  is defined.

State in your own words what it means for a function to be continuous on the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] ( a,b ).

## Algebraic

For the following exercises, determine why the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is discontinuous at a given point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics>[/itex]on the graph. State which condition fails.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mtext> </mtext><mo>|</mo><mtext> </mtext><mi>x</mi><mo>+</mo><mn>3</mn><mtext> </mtext><mo>|</mo><mo>,</mo><mi>a</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></annotation-xml></semantics>[/itex]

discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] a=−3;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(−3)  does not exist

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mtext> </mtext><mo>|</mo><mtext> </mtext><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn><mtext> </mtext><mo>|</mo><mo>,</mo><mi>a</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] 2 5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] x 2 −16 x+4 ,a=−4

removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] a=−4;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(−4)  is not defined

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] x 2 −16x x ,a=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ x,   x≠3 2x,x=3  a=3

Discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] a=3;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→3 f(x)=3 , but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(3)=6 , which is not equal to the limit.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ 5,  x≠0 3,  x=0   a=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ 1 2−x , x≠2 3, x=2   a=2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics>[/itex] x→2 f(x)  does not exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ 1 x+6 , x=−6 x 2 , x≠−6   a=−6

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ 3+x, x<1 x, x=1 x 2 , x>1     a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics>[/itex] x→ 1 − f(x)=4; lim x→ 1 + f(x)=1. Therefore,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→1 f(x)  does not exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ 3−x, x<1 x, x=1 2 x 2 , x>1     a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ 3+2x, x<1 x, x=1 − x 2 , x>1     a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics>[/itex] x→ 1 − f(x)=5≠ lim x→ 1 + f(x)=−1. Thus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→1 f(x)  does not exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ x 2 , x<−2 2x+1, x=−2 x 3 , x>−2     a=−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ x 2 −9 x+3 , x<−3 x−9, x=−3 1 x , x>−3     a=−3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics>[/itex] x→− 3 − f(x)=−6,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→− 3 + f(x)=− 1 3

Therefore,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→−3 f(x)  does not exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )={ x 2 −9 x+3 , x<−3 x−9, x=−3 −6, x>−3     a=3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= x 2 −4 x−2 ,  a=2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] 2 )  is not defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= 25− x 2 x 2 −10x+25 ,  a=5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= x 3 −9x x 2 +11x+24 ,  a=−3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] −3 )  is not defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= x 3 −27 x 2 −3x ,  a=3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics>[/itex] x |x| ,  a=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] 0 )  is not defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= 2| x+2 | x+2 ,  a=−2

For the following exercises, determine whether or not the given function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics>[/itex]is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= x 3 −2x−15

Continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] (−∞,∞)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= x 2 −2x−15 x−5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )=2⋅ 3 x+4

Continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] (−∞,∞)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )=−sin( 3x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= | x−2 | x 2 −2x

Discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )=tan( x )+2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )=2x+ 5 x

Discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= log 2 ( x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mtext> </mtext><msup/></mrow></annotation-xml></semantics>[/itex] x 2

Continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] (0,∞)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= e 2x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt/></mrow></annotation-xml></semantics>[/itex] x−4

Continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] [4,∞)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )=sec( x )−3.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] x )= x 2 +sin( x )

Continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] (−∞,∞).

Determine the values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>b</mi><mtext> </mtext></annotation-xml></semantics>[/itex]and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>c</mi><mtext> </mtext></annotation-xml></semantics>[/itex]such that the following function is continuous on the entire real number line.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mrow/></mrow></mrow></annotation-xml></semantics>[/itex] { x+1, 1<x<3 x 2 +bx+c, | x−2 |≥1 }

## Graphical

For the following exercises, refer to [link]. Each square represents one square unit. For each value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mo>,</mo></annotation-xml></semantics>[/itex] determine which of the three conditions of continuity are satisfied at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  and which are not.

<figure class="small" id="CNX_Precalc_Figure_12_03_201"></figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></annotation-xml></semantics>[/itex]

1, but not 2 or 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow></annotation-xml></semantics>[/itex]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics>[/itex]

1 and 2, but not 3

For the following exercises, use a graphing utility to graph the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=sin( 12π x )  as in [link]. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.

<figure class="small" id="CNX_Precalc_Figure_12_03_202"></figure>

Which conditions for continuity fail at the point of discontinuity?

Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(0).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics>[/itex] 0 )  is undefined.

Solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)=0.

What is the domain of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics>[/itex]

For the following exercises, consider the function shown in [link].

<figure class="small" id="CNX_Precalc_Figure_12_03_203"></figure>

At what x-coordinates is the function discontinuous?

What condition of continuity is violated at these points?

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=−1,the limit does not exist. At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=1,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( 1 )  does not exist.

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=2, there appears to be a vertical asymptote, and the limit does not exist.

Consider the function shown in [link]. At what x-coordinates is the function discontinuous? What condition(s) of continuity were violated?

<figure class="small" id="CNX_Precalc_Figure_12_03_204"></figure>

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=−7  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=1.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msup><mi>x</mi></msup></mrow></mfrac></mrow></annotation-xml></semantics>[/itex] 3 +6 x 2 −7x ( x+7 )( x−1 )

The function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)= x 3 −1 x−1  is graphed in [link]. It appears to be continuous on the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] [ −3,3 ], but there is an x-value on that interval at which the function is discontinuous. Determine the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.

<figure class="small" id="CNX_Precalc_Figure_12_03_205"></figure>

Find the limit<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→1 f(x)  and determine if the following function is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=1:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mi>x</mi><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics>[/itex] x 2 +4 x≠1 2 x=1

The function is discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=1  because the limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics>[/itex]approaches 1 is 5 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( 1 )=2.

The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f(x)= sin(2x) x  is shown in [link]. Is the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=0?  Why or why not?

<figure class="medium" id="CNX_Precalc_Figure_12_03_206"></figure>

## Glossary

continuous function
a function that has no holes or breaks in its graph
discontinuous function
a function that is not continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a
jump discontinuity
a point of discontinuity in a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] x=a  where both the left and right-hand limits exist, but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] lim x→ a − f(x)≠ lim x→ a + f(x)
removable discontinuity
a point of discontinuity in a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics>[/itex] f( x )  where the function is discontinuous, but can be redefined to make it continuous