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h:f(x+h)=f(x) for some x is lebesgue measurable

Source: VJIMC 2007 2.3

June 24, 2021
real analysisset theoryfunction

Problem Statement

Let f:[0,1]→Rf:[0,1]\to\mathbb R be a continuous function such that f(0)=f(1)=0f(0)=f(1)=0. Prove that the set A:={h∈[0,1]:f(x+h)=f(x) for some x∈[0,1]}A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}is Lebesgue measureable and has Lebesgue measure at least 12\frac12.
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