= Solution
Fix $y\in E$, and take $t>0$ sufficiently small that $x+ty\in U$. For $0<s<t$, <convexity> along the line segment gives
$$
\frac{f(x+sy)-f(x)}s\leq\frac{f(x+ty)-f(x)}t.
$$
Thus the positive secant slopes are nondecreasing in their parameter, and have a limit as the parameter decreases to zero. The local <Lipschitz continuity> from part (a) bounds their absolute values by $L\|y\|$, where $L=M/r$. Hence the <directional derivative> exists as a finite real number and
$$
\boxed{|D_yf(x)|\leq L\|y\|.}
$$
For $y=0$ it is zero.
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