Is there any example satisfying the following:
$f$ is a measurable function on $\mathbb{R}^n$ with lebesgue measure $\lambda$. For any subset $N\subseteq\mathbb{R}^N$ with $\lambda(N)=0$, $f|_{\mathbb{R}^N\setminus N}$ is not a continuous function?
2026-04-01 04:58:51.1775019531
example concerning Lusin's theorem
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My example is to take $f=\chi_X$. Now chose $X$ such that $X$ and $\mathbb{R}\setminus X$ are dense and have positive measure in every interval. This is easy to arrange using well known constructions.