Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 10 / 1 / b / 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 givesThus 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 $L\Vert y\Vert$, where $L=M/r$. Hence the directional derivative exists as a finite real number andFor $y=0$ it is zero.