Can someone provide an example of a discontinuous Young function

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.

• Convex functions on $\Bbb R$ (or on any open interval) are always continuous. Aug 5 at 17:20