= Directional derivative of a convex function is sublinear
{title2=$p_x(y)=\lim_{t\downarrow0}[f(x+ty)-f(x)]/t$}
For a <locally Lipschitz function> that is <convex> near $x$, increasing secant slopes have a finite right limit. Positive rescaling of $t$ gives positive homogeneity in $y$. The <convexity> bound $f(x+t(y+z))\leq[f(x+2ty)+f(x+2tz)]/2$ gives subadditivity after subtracting $f(x)$, dividing by $t$ and taking the limit. Thus $p_x$ is a finite <sublinear functional>, although it need not be a <linear functional>.
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