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f(x+y) >= f(x)+ f(y), prove f(x) \le 2x for all x \in [0,1]

Source: Switzerland - Swiss TST 2001 p6

February 18, 2020
functionFunctional inequalityinequalitiesalgebra

Problem Statement

A function f:[0,1]→Rf : [0,1] \to R has the following properties: (a) f(x)≥0f(x) \ge 0 for 0<x<10 < x < 1, (b) f(1)=1f(1) = 1, (c) f(x+y)≥f(x)+f(y)f(x+y) \ge f(x)+ f(y) whenever x,y,x+y∈[0,1]x,y,x+y \in [0,1]. Prove that f(x)≤2xf(x) \le 2x for all x∈[0,1]x \in [0,1].
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