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IMC 2011 Day 1 Problem 1

Source:

July 30, 2011

LaTeXtopologyreal analysisreal analysis unsolved

Problem Statement

Let

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

be a continuous function. A point

x

is called a shadow point if there exists a point

y\in \mathbb{R}

with

y>x

such that

f(y)>f(x).

Let

a<b

be real numbers and suppose that

\bullet

all the points of the open interval

I=(a,b)

are shadow points;

\bullet

a

and

b

are not shadow points. Prove that a)

f(x)\leq f(b)

for all

a<x<b;

b)

f(a)=f(b).

Proposed by José Luis Díaz-Barrero, Barcelona

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