Lebl-Basic-Analysis-I · Problem 4.2.4

The Derivative

Mathematics · derivative-sequence-approximation

Problem

Suppose

f : [a,b] \to \mathbb{R}

is differentiable and

c \in [a,b]

. Show there exists a sequence

\{x_n\}_{n=1}^\infty

converging to

c

x_n \neq c

for all

n

, such that

f'(c) = \lim_{n \to \infty} f'(x_n).

Do note this does not imply that

f'

is continuous (why?).