Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 10 / 1 / c / Solution

Write $p(y)=D_{y}f(x)$. Rescaling the parameter in the directional derivative gives $p(ay)=ap(y)$ for $a>0$, and $p(0)=0$ handles $a=0$. For sufficiently small $t>0$, convexity givesSubtract $f(x)$, divide by $t$, and let $t\downarrow 0$. The finite limits from part (b) giveThese are exactly the defining conditions of a sublinear functional. In particular, a directional derivative of a convex function is sublinear; it need not be linear.