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 $f:\mathbb{R}^d\rightarrow \mathbb{R}$?



Empty interior lack of minima
I'm always afraid of confusing lsc and usc, but what about $d=1$ and $U=\mathbb R\setminus\mathbb Q$ and $f(0)=0$ and $f(x) =1$ for $x\neq 0$?


