Lower semi-continuous function $f : C \rightarrow \mathbb{R} $ such that $C\subset \mathbb{R}^2$, $C$ is compact and convex

Question

is it possible to come up with a compact and convex set $C$ $\subset \mathbb{R}^2$, and a function $f: C \rightarrow \mathbb{R}$ where $f$ will be semi lower-continuous?

is there a way to show that, the $f$ we came up with is not continuous?

my idea was $C = \{(x,y)|x^2+y^2 \leq 1\}$ and $ f = \cases{0 & if $|x| \le \frac{1}{2}$ \cr 1 & if $|x| > \frac{1}{2}$} $

does it affect that values of x only range from -1 to 1, for the lower semi-continuity?

Thanks!