I'm not sure these two statement are not same thing.
$$ "f equals a continuous function a.e." & "f is continuous a.e."$$
The concept is too much abstract, so I wanna find some counter examples.
a function $f$ and a continuous function $g$ s.t. $f=g$ a.e and $f$ is not continuous a.e.
a function $f$ continuous a.e. s.t. there exists no continuous function $g$ with $f=g$ a.e.
In addition I wanna find a Riemann integrable function which has an uncountable set of discontinuities.
Is there some nice example to understand those concepts, note that please.