About the measurability of unbounded function on interval

Question

I was asked if a real function $f$ on $[a,b]\subseteq \mathbb{R}$ with countably many points of discontinuity is $\mathcal{B}_{\mathbb{R}}$- measurable.

If $f$ were bounded, I know it's Riemann/Lebesgue integrable, meaning that it is measurable.

1. What if $f$ were unbounded?

Thanks!

Answer 1

If you know the answer for the bounded case, then the unbounded case can be easily resolved by considering $g(x) = \arctan(f(x))$ instead. ($f$ is measurable if and only if $g$ is measurable.)