real analysis - Logical Relations Between Three Statements about Continuous Functions

(a) $f$ is continuous almost everywhere

(b) there exists a continuous function $g$ such that $f = g$ almost everywhere (on every set of non-zero measure)

(c) $f$ is nearly a continuous function (continuous everywhere but a set of measure epsilon, for small positive epsilon)

Here $f,g$ are functions from a measurable compact domain $D \subset \mathbb{R}$ to the whole real line, our measure is the Lebesgue measure, and we use the standard topology on $\mathbb{R}$.

Clearly (c) does not imply (a) as (a) is a stronger statement. Similarly, (a) implies (c).

It seems that if $f$ is a continuous function almost everywhere then we can set up a function $h$ that is continuous everywhere and set $h=f$ so (a) $\Rightarrow$ (b).

If there exist functions $g,f$ as in (b) then $f$ is nearly continuous so (b) implies (c).

Are these relations correct? Am I missing anything?