Definitions and some motivation:
Thomae’s function, also known as the raindrop function has several curious properties. One of which is the following - both its discontinuity set and continuity set are dense in $[0, 1]$, and also every discontinuity point is a removable discontinuity.
Can this happen in a “measure theoretic“ sense as well? In the sense that all the above conditions hold modulo null sets, in a way that is made precise below.
Question set up:
For a measurable function $f:\mathbb [0, 1] \to \mathbb R$, we say that $x \in \mathbb R$ is a essentially removable discontinuity of $f$ if
i) $f$ is not continuous at $x$,
ii) There exists a measure zero set $N$, and a measurable function $g: \mathbb [0, 1] \to \mathbb R$ such that $g$ is continuous at $x$, and $g$ agrees with $f$ everywhere except possibly on $N$.
Note that here both $N$ and $g$ are allowed to depend on $x$.
Question: Denote by $\mathcal E$ the set of essentially removable discontinuities of $f$, and $\mathcal C$ the set of continuity points respectively. Does there exist a function f such that $[0, 1]= \mathcal E \cup \mathcal C$ and such that both $\mathcal E$ and $\mathcal C$ have nonzero measure in any open interval?
