Lebl-Basic-Analysis-I · Problem 3.4.16

Continuous Functions

Mathematics · uniform-continuity

Problem

Suppose

f : S \to \mathbb{R}

and

g : [0, \infty) \to [0, \infty)

are functions,

g

is continuous at

0

g(0) = 0

, and whenever

x

and

y

are in

S

, we have

|f(x) - f(y)| \leq g\bigl(|x-y|\bigr)

. Prove that

f

is uniformly continuous.