Directional derivative of a convex function is sublinear ($p_x(y)={\lim }_{t\downarrow 0}[f(x+ty)-f(x)]/t$)

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 $f(x+t(y+z))\le [f(x+2ty)+f(x+2tz)]/2$ 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.