= Negative part of a finite convex function is Lipschitz
{title2=$|f^-(x)-f^-(y)|\leq L|x-y|$}
For a finite <convex function> $f:\mathbb R\to\mathbb R$, its negative part $f^-=\max(-f,0)$ is globally <Lipschitz continuous>. The negative sublevel set is an interval. If bounded, local Lipschitz continuity of finite convex functions bounds the slopes on its closure. If it is a left ray, any negative slope there would force $f$ positive far to the left by the supporting-line inequality; all slopes on that ray are therefore nonnegative and bounded above near its finite endpoint. For a right ray the analogous slopes are nonpositive and bounded below. If $f$ is negative everywhere, convexity and boundedness above force it to be constant. These cases bound the slopes of $f^-$ globally, including its zero extension outside the interval. In particular, integrability of $f(Z)$ implies integrability of the negative part of every fixed translate $f(Z+x)$.
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