Let me first define the notion of a Young function:
A convex function $\Phi: \mathbb{R} \to \mathbb{R}^+$ which satisfies the following conditions
(1) $\Phi(0)=0$
(2) $\Phi(-x)=\Phi(x)$
(3) $\lim_{x\to \infty}\Phi(x)=\infty$
is called a Young function. I know that convex functions are not always continuous, but I need a convex function that satisfies all the above conditions and that is discontinuous.
