Suppose you are given a measurable set $E\subset[0,1]$ such that for any nonempty open sub-interval $I$ in $[0,1]$, both sets $E \cap I$ and $E^c \cap I$ have positive measure. Then, for the function $f:=\chi_E$, where $\chi_E$ is characteristic function, show that whenever $g(x)=f(x)$ a.e. in $x$, then $g$ must be discontinuous at every point in $[0,1]$ .
I think we can take advantage of the problem of making a measurable subset $E ⇢ [0, 1]$ such that for every sub-interval $I$, both $E \ I$ and $I - E$ have positive measure by taking a Cantor-type subset of $[0, 1]$ with positive measure and on each sub-interval of the complement of this set, and construct another such set, and so on. I don't know if I am right.
It shouldn't require any constructions. Take $x\in (0,1)$. Then for any open interval $I$ around $x$, we know that $f$ takes the value 1 on a positive measure set, and the value 0 on a positive measure set. Then because $g=f$ a.e. the same is true for $g$. So $g$ cannot be continuous at $x$.
The boundary points work more or less the same, just using the open intervals $(0,\varepsilon)$ and $(1-\varepsilon,1)$.