4.2: The Derivative Requires Continuity
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Suppose u is a function.
Iflimb→3u(b)−5b−3=4,what islimb→3u(b)?
The answer is that
limb→3u(b)=5
We reason that
for b close to 3, the numerator ofu(b)−5b−3is close to 4 times the denominator.That is,u(b)−5is close to4×(b−3).
But 4×(b−3) is also close to zero. Therefore
If b is close to 3, u(b)−5 is close to zero and u(b) is close to 5.
The general question we address is:
Theorem 4.2.1 The Derivative Requires Continuity
If u is a function and u′(t) exists at t=a then u is continuous at t=a.
- Proof
-
In Exercise 4.2.2 you are asked to give reasons for the following steps, (i) − (v). Suppose the hypothesis of Theorem 4.2.1.
(limb→au(b))−u(a)=limb→a(u(b)−u(a))(i)=limb→au(b)−u(a)b−a×(b−a)=limb→au(b)−u(a)b−a×limb→a(b−a)(ii)=u′(a)limb→a(b−a)(iii)=0(iv)(limb→au(b))=u(a)(v)
End of proof.
A graph of a function u defined by
u(t)={0 for 20≤t<2850 for t=28100 for 28<t≤30
is shown in Figure 4.2.1A. We observed in Section 4.1 that u is not continuous at t=28. (limt→28−u(t)=0≠50=u(28).) Furthermore, u′(28) does not exist. A secant to the graph through (b,0) and (28,50) with b<28 is drawn in Figure 4.2.1B, and
for b<28,u(b)−u(28)b−28=0−50b−28=5028−b
The slope of the secant gets greater and greater as b gets close to 28.
Figure 4.2.1: A. Graph of The function u defined in Equation ???. B. Graph of u and a secant to the graph through (b,0) and (28,50).
Explore 4.2.1 Is there a line tangent to the graph of u shown in Figure 4.2.1 at the point (28,50) of the graph?
Figure 4.2.2: a. Graph of P(t)=|t| for all t. B. Graph of (P(t)−P(0))/(t−0).
Explore 4.2.2 In Explore Figure 4.2.2 is the graph of y=√|x|. Does the graph have a tangent at (0,0)? Your vote counts.
Explore Figure 4.2.2 Graph of y=√|x|.
The graph of P(t)=|t| for all t is shown in Figure 4.2.2A. P is continuous, but P′(0) does not exist.
P(b)−P(0)b−0=|b|−0b−0=|b|b={−1 for b<01 for b>0
A graph of P(b)−P(0)b−0 is shown in Figure 4.2.2B. It should be clear that
limb→0P(b)−P(0)b−0does not exist, so thatP′(0)does not exist.
Therefore, the converse of Theorem 4.2.1 is not true. Continuity does not imply that the derivative exists.
Exercises for Section 4.2, The Derivative Requires Continuity.
Exercise 4.2.1 Shown in Figure 4.2.1 is the graph of C(t)=3√t.
- Use Definition of Derivative Equation 3.3.3 to show that C′(0) does not exist.
- Is C(t) continuous?
- Is there a line tangent to the graph of C at (0,0)?
Figure for Exercise 4.2.1 Graph of \(C(t) = \sqrt[3] {t} for Exercise 4.2.1.
Exercise 4.2.2 Justify the steps (i) − (v) in Equations 4.2.4.
Exercise 4.2.3 For for the function P(t)=|t| for all t compute P′−(0) and P′+(0).
Explore 4.2.3 This problem may require extensive thought. Is there a function defined for all numbers t and continuous at every number t and for which f′−(1) does not exist?