A function $u: \mathbb{R}^n \to \mathbb{R}$ is defined to be semiconcave if there is a positive constant $c$ such that for all $x,z$ $$ u(x-z) + u(x+z) - 2u(x) \leq c |z|^2. $$ Alternatively, one defines $u$ to be semiconcave if, for some constant $c > 0$, $u(z) - c|z|^2$ is concave which would imply that $u$ has to be continuous.
I know that the two definitions are equivalent if $u$ is locally bounded or if $u$ is measurable (cf. https://shreevatsa.wordpress.com/2010/06/29/convex-continuous-jensens/). Hence, if there is a discontinuous semiconcave function, then it has to be non-measurable and blow up in at least one point...
My question: Can you give an example of a discontinuous semiconcave function? Or does the inequality above already imply continuity of $u$?
