# Supposed to be simple but seems a bit tricky

1. May 7, 2010

### shultz

Let $$f:{R^n} \to R$$ be a continuous function on some $$D \subset {R^n}$$.
Now, define
$$epi\left( f \right) = \left\{ {\left( {x,r} \right) \in {R^{n + 1}}:x \in {R^n},r \ge f\left( x \right)} \right\}$$​
I want to prove to myself (not homework) that if $$D$$ has a non-empty interior (that is, there exist interior points in $$D$$), so also $$epi\left( f \right)$$ has a non-empty interior.
I think it is supposed to be easy but I get stuck.

I started with the following simple argument. If $$D$$ has a non-empty interior, then there exist $${x_0} \in D$$ and $$r > 0$$ in $$R$$ such that $$B\left( {{x_0},r} \right) \subset D$$ (that is, the ball with radius $$r$$ around $${x_0}$$ is contained in $$D$$). Now, because $$f$$ is continuous, there also exist $$0 < {r_2} < r$$ in $$R$$ such that $$f\left( x \right) < f\left( {{x_0}} \right) + 1$$ for all $$x \in B\left( {{x_0},{r_2}} \right)$$.
Now, I want to show that $$\left( {{x_0},f\left( {{x_0}} \right) + 1} \right) \in epi\left( f \right)$$ but I do not have a good idea how to continue.