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CIIM 2009 Problem 5

Source:

June 9, 2016

CIIM 2009CIIMUndegraduate

Problem Statement

Let

f:\mathbb{R} \to \mathbb{R}

, such that
i) For all

a \in \mathbb{R}

and all

\epsilon > 0

, exists

\delta > 0

such that

|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.

ii) For all

b\in \mathbb{R}

and all

\epsilon > 0

, exists

x,y \in \mathbb{R}

with

b - \epsilon < x < b < y < b + \epsilon

, such that

|f(x)-f(b)|< \epsilon

and

|f(y)-f(b)| < \epsilon.

Prove that if

f(a) < d < f(d)

there exists

c

with

a < c < b

or

b < c < a

such that

f(c) = d

.

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