Suppose I have a sequence of continuous functions converging pointwise almost everywhere to a continuous function. Is it true that we have pointwise everywhere convergence?
Consider the class $[f]$ of all functions which are ae equal to $f$. You can show that each such class contains at most 1 continuous function (if $f \neq g$ at some point and both are continuous, you can find a neighborhood of that point where they're unequal). So yes, you can modify the functions on a null set, but you're no longer talking about a sequence of continuous functions. I'm not sure why you'd want to do this though. Did you have an application in mind?