Is a locally bounded function $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ also "bounded almost everywhere"?
Is the viceversa true?
Notes.
Definition of "local boundedness":
$f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ is locally bounded if for any $x \in \mathbb{R}^n$ there exists a neighborhood $A$ of $x$ such that $f(A)$ is a bounded set, that is, for some $M > 0$ we have $f(x) \leq M$ for all $x \in A$.
Definition of "almost-everywhere boundedness":
$f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ is bounded almost everywhere on $\mathbb{R}^n$ if there exist $M>0$ and a set $E \in \mathbb{R}^n$ of measure $0$ so that $f(x) \leq M$ for any $x \notin E$.
Best Answer
Consider the function $f:\mathbb{R}\to\mathbb{R}_{\geq 0}$ given by $$f(x)=\begin{cases}0 & x=0\:\mathrm{or}\: x\notin\mathbb{Q}\\1/|x| & \mathrm{otherwise}\end{cases}.$$ Is $f$ bounded almost everywhere? Is $f$ locally bounded?