Lebl-Basic-Analysis-I · Problem 4.1.11

The Derivative

Mathematics · differentiability-product

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) := f(x)g(x)

is differentiable at

c

. Hint: You cannot apply the product rule.