Basic Analysis I — Real Analysis · Problem 4.1.12

The Derivative — Jiri Lebl

Mathematics — Real Analysis · differentiability-squeeze-theorem

Problem

Suppose

f : I \to \mathbb{R}

g : I \to \mathbb{R}

, and

h : I \to \mathbb{R}

, are functions. Suppose

c \in I

is such that

f(c) = g(c) = h(c)

g

and

h

are differentiable at

c

, and

g'(c) = h'(c)

. Furthermore, suppose

h(x) \leq f(x) \leq g(x)

for all

x \in I

. Prove

f

is differentiable at

c

and

f'(c) = g'(c) = h'(c)