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

Choose $M$ with $|f|\le M$ on $U$, and fix $r=(1-\Vert x\Vert )/4>0$. We prove a local Lipschitz continuity bound, which gives more than continuity. Let $u,v\in B(x,r)$, put $d=\Vert v-u\Vert$, and suppose $d>0$. The point $w=u+(2r/d)(v-u)$ lies in $B(x,3r)\subset U$, while $d<2r$ andBy convexity, $f(v)-f(u)\le [d/(2r)](f(w)-f(u))\le Md/r$. Interchanging $u,v$ givesThe case $d=0$ is immediate. Every point has such a neighborhood, so $f$ is locally Lipschitz and therefore continuous throughout $U$. This is the bounded convex functions are locally Lipschitz on an open ball argument; no finite-dimensional assumption is needed.