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A condition which implies linearity

Source: Miklós Schweitzer 2013, P8

July 12, 2014
functionreal analysisreal analysis unsolved

Problem Statement

Let f:R→R{f : \Bbb{R} \rightarrow \Bbb{R}} be a continuous and strictly increasing function for which f−1(f(x)+f(y)2)(f(x)+f(y))=(x+y)f(x+y2) \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) for all x,y∈R(f−1{x,y \in \Bbb{R}} ({f^{-1}} denotes the inverse of f){f}). Prove that there exist real constants a≠0{a \neq 0} and b{b} such that f(x)=ax+b{f(x)=ax+b} for all x∈R.{x \in \Bbb{R}}.
Proposed by Zoltán Daróczy
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