Suppose that $U \subseteq \mathbb{R}^d$, and satsifies

- $U$ is dense in $\mathbb{R}^d$,
- U has empty interior,

Then is it possible that $$ \inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x), $$ for some (fixed) lsc function $f:\mathbb{R}^d\rightarrow \mathbb{R}$?