SCHWEITZER 2013 Problem 9

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