Basic Analysis I — Real Analysis · Problem 3.6.10

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-functions-continuity

Problem

a) Let

S \subset \mathbb{R}

be a subset. If

f : S \to \mathbb{R}

is increasing and bounded, then show that there exists an increasing

F : \mathbb{R} \to \mathbb{R}

such that

f(x) = F(x)

for all

x \in S

. b) Find an example of a strictly increasing bounded

f : S \to \mathbb{R}

such that an increasing

F

as above is never strictly increasing.