Closedness of marginal epigraph?

Question

Given closed convex function $f(x,y):\Bbb R^n \times \Bbb R^m \to \Bbb R$, this means the epigraph of $f$ is a closed set. Will the epigraph of the marginal function $f_{x_0}(y)=f(x_0,y)$ always be closed for any $x_0$?

I feel that this might not be true, because projection of a closed set might not be closed (e.g. the epigraph of $1/x,x\ge $ is a closed set, but its projection onto x-axis is open).

But I have trouble finding a counterexample that marginal epigraph could be open. Any help is appreciated. Thanks!

Answer 1

It is closed, If question has been stated correctly ! First note that when you write $f:\Bbb R^n \times \Bbb R^m \to \Bbb R$ this means $dom(f) = \Bbb R^n \times \Bbb R^m$. and since $f$ is convex on whole space it is continuous so is automatically closed (you don't need make this assumption)! In particular $f_{x_0} : \Bbb R \to \Bbb R$ with $f_{x_0}(y)=f(x_0,y)$ is convex and so it is closed!