PomeA7E9B5A114CATheorem 21.11.20.A7E9B5A114CA Let $C\subseteq N$ be an open subset. Let $(f_ i)_{i>0}$ be a sequence of real-valued convex functions on $C$. Suppose that the sequence converges on a dense subset of $C$ and the limit is finite. Then the limit
exists for all $x\in C$ and is convex on $C$. Moreover, the sequence $(f_ i)_ i$ converges uniformly to $f$ on each compact subset of $C$.
Proof. On every open convex ball relatively compact in $C$, dense convergence for finite convex functions applies to the restriction of the given dense set and supplies a finite convex local limit with locally uniform convergence. Local limits agree on overlaps, because they are pointwise limits of the same sequence. They therefore define a finite function $f$ on $C$.
Every compact subset of $C$ is covered by finitely many smaller such balls. Taking the maximum of the finitely many uniform error bounds proves uniform convergence on that compact set. Passing to the limit in each convexity inequality proves convexity wherever the corresponding segment lies in $C$; in the usual convention that the domain of a convex function is convex, this is convexity on $C$. In particular the limiting values at the original dense set determine $f$ uniquely.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.