Basic Analysis I — Real Analysis · Problem 3.6.13

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-functions-continuity

Problem

Suppose

f : [0, 1] \to (0, 1)

is increasing. Show that for every

\epsilon > 0

, there exists a strictly increasing

g : [0, 1] \to (0, 1)

such that

g(0) = f(0)

f(x) \leq g(x)

for all

x

, and

g(1) - f(1) < \epsilon