Does $f(\frac{x+y}{2})\leqslant \frac{f(x)+f(y)}{2}$ imply continuity?
Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex