Chapter 3 | Derivatives
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however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.
Theorem 3.1: Differentiability Implies Continuity Let f ( x ) be a function and a be in its domain. If f ( x ) is differentiable at a , then f is continuous at a .
Proof If f ( x ) is differentiable at a , then f ′( a ) exists and
f ′( a ) = lim x → a f ( x )− f ( a ) x − a . We want to show that f ( x ) is continuous at a by showing that lim x → a
f ( x ) = f ( a ). Thus,
f ( x ) = lim x → a ⎛
⎝ f ( x )− f ( a )+ f ( a ) ⎞ ⎠
lim x → a
⎛ ⎝ f ( x )− f ( a ) x − a f ( x )− f ( a ) x − a
· ( x − a )+ f ( a ) ⎞ ⎠
Multiply and divide f ( x )− f ( a )by x − a .
= lim x → a
⎛ ⎝ lim x → a
⎞ ⎠ ·
⎛ ⎝ lim x → a (
⎞ ⎠ + lim x → a
x − a )
=
f ( a )
= f ′( a ) ·0+ f ( a ) = f ( a ). Therefore, since f ( a ) is defined and lim x → a
f ( x ) = f ( a ), we conclude that f is continuous at a .
□ We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function f ( x ) = | x |. This function is continuous everywhere; however, f ′(0) is undefined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further. For f ( x ) = | x |, f ′(0) = lim x →0 f ( x )− f (0) x −0 = lim x →0 | x | − |0| x −0 = lim x →0 | x | x . This limit does not exist because lim x →0 − | x | x =−1and lim x →0 + | x | x =1. See Figure 3.14 .
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