= Continuous supporting functional for a locally bounded convex function
{title2=$f(x+z)\geq f(x)+l(z),\quad l\in E'$}
A real <convex function> locally bounded on an open <convex set> has a continuous supporting <linear functional> at every interior point. Apply the dominated <Hahn-Banach theorem> to its sublinear <directional derivative> $p_x$ to obtain $l\leq p_x$. Increasing secant slopes imply $p_x(z)\leq f(x+z)-f(x)$ whenever the endpoint is in the domain. A local <Lipschitz continuity> bound gives both $l(y)\leq L\|y\|$ and $-l(y)=l(-y)\leq L\|y\|$, proving continuity.
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