Basic Analysis I — Real Analysis · Problem 7.5.16

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions-metric-spaces

Problem

Let

(X,d_X)

and

(Y,d_Y)

be metric spaces,

S \subset X

p \in X

a cluster point of

S

, and let

f: S \to Y

be a function. Prove that

f: S \to Y

is continuous at

p

if and only if

\lim_{x \to p} f(x) = f(p).