Basic Analysis I — Real Analysis · Problem 7.5.1

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

Consider

\mathbb{N} \subset \mathbb{R}

with the standard metric. Let

(X, d)

be a metric space and

f : X \to \mathbb{N}

a continuous function. a) Prove that if

X

is connected, then

f

is constant (the range of

f

is a single value). b) Find an example where

X

is disconnected and

f

is not constant.