This is on the construction of a measurable function $f$ on $[0,1]$ such that every function $g$ that differs from $f$ only ona set of measure zero is discontinuous at every point.
The exercise is #37 from pp. 45 and it asks the following:
(a) Construct a measurable set $E \subset [0,1]$ such that for any non-empty opensub-interval $I$ in $[0,1]$, both sets $E \cap I$ and $E^{c} \cap I$ have positive measure.
(b) Show that $f = \chi_{E}$ has the property that whenever $g(x) = f(x)$ almost everywhere, then $g$ must be discontinuous at every point in $[0,1]$.
While I think I got (a) using the hint to consider Cantor-like sets, I am stuck at (b); thanks in advance for any help.