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2.4: Continuity

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Continuity

A function that is "friendly" and doesn’t have any breaks or jumps in it is called continuous. More formally,

Continuity at a Point

A function f is continuous at x=a if and only if limxaf(x)=f(a).

The graph below illustrates some of the different ways a function can behave at and near a point, and the table contains some numerical information about the function and its behavior.

graph
a f(a) limxaf(x)
1 2 2
2 1 2
3 2 Does not exist (DNE)
4 Undefined 2

Based on the information in the table, we can conclude that f is continuous at 1 since limx1f(x)=2=f(1).

We can also conclude from the information in the table that f is not continuous at 2 or 3 or 4, because limx2f(x)f(2), limx3f(x)f(3), and limx4f(x)f(4).

The behaviors at x=2 and x=4 exhibit a hole in the graph, sometimes called a removable discontinuity, since the graph could be made continuous by changing the value of a single point. The behavior at x=3 is called a jump discontinuity, since the graph jumps between two values.

So which functions are continuous? It turns out pretty much every function you’ve studied is continuous where it is defined: polynomial, radical, rational, exponential, and logarithmic functions are all continuous where they are defined. Moreover, any combination of continuous functions is also continuous.

This is helpful, because the definition of continuity says that for a continuous function, limxaf(x)=f(a). That means for a continuous function, we can find the limit by direct substitution (evaluating the function) if the function is continuous at a.

Example 2.4.5

Evaluate using continuity, if possible:

  1. limx2x34x
  2. limx2x4x+3
  3. limx2x4x2

Solution

  1. The given function is polynomial, and is defined for all values of x, so we can find the limit by direct substitution:limx2x34x=234(2)=0.
  2. The given function is rational. It is not defined at x=3, but we are taking the limit as x approaches 2, and the function is defined at that point, so we can use direct substitution:limx2x4x+3=242+3=25.
  3. This function is not defined at x=2, and so is not continuous at x=2. We cannot use direct substitution.

This page titled 2.4: Continuity is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Shana Calaway, Dale Hoffman, & David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.

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