MathDB

Let

{f : \Bbb{R} \rightarrow \Bbb{R}}

be a continuous and strictly increasing function for which

\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 \in \Bbb{R}} ({f^{-1}}

denotes the inverse of

{f})

. Prove that there exist real constants

{a \neq 0}

and

{b}

such that

{f(x)=ax+b}

for all

{x \in \Bbb{R}}.

Proposed by Zoltán Daróczy

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