I'm trying to prove that if $f$ is measurable and finite a.e. on $[a,b]$, than given $\epsilon>0$, there exists a continuous $g$ on $[a,b]$ s.t. the measure of the set $\{x: f(x) \text{ is different from } g(x)\}$ is less than $\epsilon$. (Weeden, Zygmund: measure and integral 2nd exercise 20 of ch4 p.78)
Intuitively, since I'm working on a real line, I guess the removable and jump discontinuities can be easily handled, but I don't know how to write down this rigorously.
And for the essential discontinuities, i'm completely lost. For example, let $f(x)=\frac 1x$ and take the interval $[0,1]$. Then clearly it is measurable and finite a.e. on that domain, but I can't find a way to construct $g$ to be continuous at $0$.
Finally, I'm not even sure (and can't prove) that those three cases cover all. I know that the first two things are called the first kind and the essential one is called the second kind but didn't learn this deep.