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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.

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

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