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Prove that there is a function

{f: (0,\infty) \rightarrow (0,\infty)}

which is nowhere continuous and for all

{x,y \in (0,\infty)}

and any rational

{\alpha}

we have

\displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}.

Is there such a function if instead the above relation holds for every

{x,y \in (0,\infty)}

and for every irrational

{\alpha}?

Proposed by Maksa Gyula and Zsolt Páles

Problem Statement