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max of |f(x) - f(y)|, when f:[0,1]\to R, f(0)=0,f(1)=1,|f(x) -f(y)|<= c|x - y|

Source: Rioplatense Olympiad 2016 level 3 P4

September 5, 2018
functionalgebramaximumFunctional inequality

Problem Statement

Let c>1c > 1 be a real number. A function f:[0,1]→Rf: [0 ,1 ] \to R is called c-friendly if f(0)=0,f(1)=1f(0) = 0, f(1) = 1 and ∣f(x)−f(y)∣≤c∣x−y∣|f(x) -f(y)| \le c|x - y| for all the numbers x,y∈[0,1]x ,y \in [0,1]. Find the maximum of the expression ∣f(x)−f(y)∣|f(x) - f(y)| for all c-friendly functions ff and for all the numbers x,y∈[0,1]x,y \in [0,1].
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