# Empty interior lack of minima

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

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 $$x\neq 0$$?
• Cool, I came up with a slightly reduced version of this concentrated at a single point in $\mathbb{R}^d-U$. Thanks :) – AIM_BLB Feb 8 at 18:26