Basic Analysis I — Real Analysis · Problem 3.1.9

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · limits-of-functions

Problem

Let

c_1

be a cluster point of

A \subset \mathbb{R}

and

c_2

be a cluster point of

B \subset \mathbb{R}

. Suppose

f : A \to B

and

g : B \to \mathbb{R}

are functions such that

f(x) \to c_2

x \to c_1

and

g(y) \to L

y \to c_2

. If

c_2 \in B

, also suppose that

g(c_2) = L

. Let

h(x) := g\bigl(f(x)\bigr)

and show

h(x) \to L

x \to c_1

. Hint: Note that

f(x)

could equal

c_2

for many

x \in A

, see also Exercise 3.1.14.