Basic Analysis I — Real Analysis · Problem 3.6.9

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-functions-continuity

Problem

Consider a monotone function

f : I \to \mathbb{R}

on an interval

I

. Prove that there exists a function

g : I \to \mathbb{R}

such that

\displaystyle\lim_{x \to c^-} g(x) = g(c)

for all

c

I

except the smaller (left) endpoint of

I

, and such that

g(x) = f(x)

for all but countably many

x \in I