Is $f$ bounded a.e. if $f$ is finite?

Question

I'm trying to figure out whether the following statement is true:

If $f$ is a measurable function on $E$ and is finite almost everywhere on $E$ (i.e. $|f(x)|<\infty$) then $f$ is bounded a.e.

I used this statement in a proof but it was marked incorrect. I'm having trouble producing a counterexample.

Answer 1

Consider the function $f(x) = x\sin(x)$ (or even $f(x) = x$).

Answer 2

$1/x$ on $(0,1]$ is finite a.e.