TITLE: Finite at every point but unbounded on every interval
QUESTION [7 upvotes]: Is is possible that a function $f$ is finite at every point but unbounded on every interval?
What if f is measurable?

REPLY [4 votes]: Asaf's example is very nice due to its additional property. An easier example (without worrying about the intermediate value property) is to let $f(x)$ be whatever you want if $x$ is irrational, it doesn't matter, while if $x=p/q$ is rational, with $\gcd(p,q)=1$ and $q>0$, then we set $f(x)=q$.