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Convex function and functional equation

Source: Romanian Nationals RMO 2005 - grade 11, problem 4

March 31, 2005
functionlimitalgebra proposedalgebra

Problem Statement

Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a convex function. a) Prove that ff is continous; b) Prove that there exists an unique function g:[0,∞)→Rg:[0,\infty)\to\mathbb{R} such that for all x≥0x\geq 0 we have f(x+g(x))=f(g(x))−g(x). f(x+g(x)) = f(g(x)) - g(x) .
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