A function $f$ is called a Baire 1 function if it is a pointwise limit of a sequence of continuous functions. I was wondering if $f$ is almost everywhere pointwise convergence of continuous functions, is true that $f$ is a continuous function?
For example $f_n(x) = \frac{1}{1+(nx)^2}$ converges to $g$ everywhere, where $g(x) = 0$ if $x \neq 0$ and $g(x) = 1$ if $x=0$. Further, this sequence converges to $h(x) = 0$ almost everywhere. Evidently, the function $g$ is Baire 1 but not continuous and function $h$ is a continuous function.
Consider functions on the interval $[0,2].$
Now take $f(x)= \chi_{[1,2]},$ where $\chi_A$ is the indicator function of the set $A$. If you define the sequence of continuous functions: $$f_n(x) : = \begin{cases} x^n & x \le 1 \\ 1 & x \ge 1 \end{cases}$$
Then you see that $f_n \to f$ pointwise.
Furtheremore it is not possible to modify $f$ on a nullset, so that it becomes continuous (why is that?).