Basic Analysis I — Real Analysis · Problem 4.1.11

The Derivative — Jiri Lebl

Mathematics — Real Analysis · differentiability-product-bounded

Problem

Suppose

f : I \to \mathbb{R}

is bounded,

g : I \to \mathbb{R}

is differentiable at

c \in I

, and

g(c) = g'(c) = 0

. Show that

h(x) \mathrel{\mathop:}= f(x)g(x)

is differentiable at

c

. Hint: You cannot apply the product rule.