1
$\begingroup$

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.

1 Answers 1

2

No; in fact $f$ is continuous 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. Left-continuity implies that the left limit is the function value and right-continuity implies that the right limit is the function value, so both together imply that the left limit, right limit and function value coincide, which means that the function is continuous.

By contrast, a regulated function can have arbitrary values at isolated points, since that doesn't affect the left or right limit at any point.

  • 0
    Thanks; I feel silly for not thinking of this myself.2012-02-28