Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.2 Working with Derivatives - 3.2 Exercises: 4


It doesn't have to be differentiable at $a$.

Work Step by Step

The counterexample would be the function $$f(x)=\left\{_{x,\quad x<1}^{-x+2,\quad x\geq1}\right.$$ This function at $x=1$ has the derivative equal to $1$ from the left side but equal to $-1$ from the right side so the derivative at $x=1$ doesn't exist even though the function is continuous at $x=1$. The graph of this function is on the figure below.
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