In the context of measurable functions from the reals to the reals:
Let's say a function is essentially continuous at a domain point $c$ iff
$$\exists L, \forall \epsilon, \exists \delta, B_\delta (c) \subseteq \textrm{Cl}(f^{-1}(B_\epsilon(L)))$$
Meaning that there exists some point in the codomain such that the closure of the inverse image of any open ball around that point is a superset of an open ball around the domain point, $c$.
Which means it breaks essential continuity if:
$$\forall L, \exists \epsilon, \forall \delta, B_\delta (c) \nsubseteq \textrm{Cl}(f^{-1}(B_\epsilon(L)))$$
I'm wondering about the size of this set of discontinuities. I'm pretty sure it can at least be countable. But I'm wondering if it can be uncountable or have Lebesgue Measure $> 0$.