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Equal values of continuous function

Source: Kvant Magazine No. 1 2020, 50th Anniversary, M413

March 10, 2023
real analysisfunctioncontinuity

Problem Statement

Determine the positive numbers aa{} for which the following statement true: for any function f:[0,1]→Rf:[0,1]\to\mathbb{R} which is continuous at each point of this interval and for which f(0)=f(1)=0f(0)=f(1)=0, the equation f(x+a)−f(x)=0f(x+a)-f(x)=0 has at least one solution.
Proposed by I. Yaglom
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