= Solution
Write $p(y)=D_yf(x)$. 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> gives
$$
f(x+t(y+z))\leq\tfrac12f(x+2ty)+\tfrac12f(x+2tz).
$$
Subtract $f(x)$, divide by $t$, and let $t\downarrow0$. The finite limits from part (b) give
$$
\boxed{p(y+z)\leq p(y)+p(z),\qquad p(ay)=ap(y)\quad(a\geq0).}
$$
These 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.
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