Basic Analysis I — Real Analysis · Problem 3.2.18

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

Suppose

f : [-1, 1] \to \mathbb{R}

is a function continuous at all

x \in [-1, 1] \setminus \{0\}

. Show that for every

\epsilon

such that

0 < \epsilon < 1

, there exists a function

g : [-1, 1] \to \mathbb{R}

continuous on all of

[-1, 1]

, such that

f(x) = g(x)

for all

x \in [-1, -\epsilon] \cup [\epsilon, 1]

, and

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

for all

x \in [-1, 1]