f is continuous if and only if epi(f) is closed

Question

Let $epi(f)=\{(x,\alpha)\in R^{n+1}| f(x)\le\alpha\}$

Prove or disprove:

f is continuous if and only if $epi(f)$ is closed. Can someone give me a hint?

This is not true, if $f = 1_{(0,1)}$ which is lower semicontinuous then the epigraph is closed. — copper.hat, May 02 '17 at 19:37

score 2 · Accepted Answer · answered May 02 '17 at 19:28

2

Let $f$ be continuous and $(x_n,y_n)\in\text{epi}(f)$ converges. What about its limit?

Let the epigraph be closed and $x_n\to x$. Observe that $\bigl(x_n,f(x_n)\bigr),\,\bigl(x,f(x)\bigr)\in\text{epi}(f)$. What follows by closedness?

answered May 02 '17 at 19:28

szw1710

1 Answers1