Basic Analysis I — Real Analysis · Problem 3.4.16

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · 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

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

. Prove that

f

is uniformly continuous.