MathDB

Increasing function implies existence of non-increasing one

Source: VJIMC 2017, Category II, Problem 2

April 2, 2017

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Problem Statement

Prove or disprove the following statement. If

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

is an increasing function and satisfies

g(x) > x

for all

x \in (0,1)

, then there exists a continuous function

f:(0,1) \to \mathbb{R}

satisfying

f(x) < f(g(x))

for all

x \in (0,1)

, but

f

is not an increasing function.