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Putnam 2014 B6

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December 8, 2014

Putnamfunctionslopeintegrationcollege contestsreal analysisPutnam 2014

Problem Statement

Let

f:[0,1]\to\mathbb{R}

be a function for which there exists a constant

K>0

such that

|f(x)-f(y)|\le K|x-y|

for all

x,y\in [0,1].

Suppose also that for each rational number

r\in [0,1],

there exist integers

a

and

b

such that

f(r)=a+br.

Prove that there exist finitely many intervals

I_1,\dots,I_n

such that

f

is a linear function on each

I_i

and

[0,1]=\bigcup_{i=1}^nI_i.

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