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

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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 $x\neq 0$?

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    $\begingroup$ Yes this is lsc. $\endgroup$ – YCor Feb 8 at 18:14
  • $\begingroup$ Cool, I came up with a slightly reduced version of this concentrated at a single point in $\mathbb{R}^d-U$. Thanks :) $\endgroup$ – AIM_BLB Feb 8 at 18:26

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