As was established in the answer to my very first question on MSE, a regulated function or jump continuous function, is a function $f:[a,b] \rightarrow \mathbb{R}$ such that (1) The limit of $f$ as $x$ approaches $a$ from above exists (2) the limit as $x$ approaches $b$ from the left exists and (3) for any point $p$ in the interior of $[a,b]$ both the left and right limits as $x$ approaches $p$ exist (but they are not necessarily equal).
My first question is, Can this definition can be cast in terms of one-sided continuity? That is, is it a true statement that $f$ is regulated if and only if it is right continuous at $a$, left continuous at $b$ and both left and right continuous at any point in the interior.
Now, if this is true, and I believe that it is, What does this say about sequential continuity of a regulated function? It is a fact that a function in a (metric) space is continuous if and only if it is sequentially continuous. But for a regulated function, we don't exactly have continuity, only one-sided continuity, and I'm not sure what one can say about the relationship between sequential continuity and one-sided continuity.