Lebl-Basic-Analysis-I · Problem 3.4.5

Continuous Functions

Mathematics · uniform-continuity

Problem

Let

A, B

be intervals. Let

f : A \to \mathbb{R}

and

g : B \to \mathbb{R}

be uniformly continuous functions such that

f(x) = g(x)

for

x \in A \cap B

. Define the function

h : A \cup B \to \mathbb{R}

h(x) := f(x)

x \in A

and

h(x) := g(x)

x \in B \setminus A

. a) Prove that if

A \cap B \neq \emptyset

, then

h

is uniformly continuous. b) Find an example where

A \cap B = \emptyset

and

h

is not even continuous.