Why do regulated functions receive so little attention in elementary analysis courses? The only place where regulated functions (= such with one-sided limits everywhere) occasionally seem to come up in elementary analysis courses is in connection with integration, yet there are clearly several places, even before continuity is discussed, where they naturally fill some gaps in theory and could be used to throw light on related concepts. Here are some arguments to justify this and hence the question in the title:


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*It is the regulated functions (and not the continuous functions) that are the righteous owner of the result "are bounded on compact invervals" and the sequential proof is virtually the same as for continuous functions.

*Regulated functions provide a natural (if a little tautological) converse to the intermediate value theorem: continuous $\Leftrightarrow$ regulated + has IVP. This characterization shows exactly what two ingredients it takes to make a continuous function.

*The main result on regulated functions (which says that they are precisely the uniform limits of step functions) nicely mingles with other analytic concepts, including uniform convergence, uniform continuity, open-cover compactness of $[a, b]$, and even countability in the form that the countable subsets are precisely the sets of discontinuity points of regulated functions.
Why do typical first analysis courses (at least in Europe) seem to avoid such facts?
 A: I think the answer ultimately lies within the context of real analysis in a mathematics curriculum. First and second years take proof based real analysis classes to become acquainted with analysis as a subject and learn proof based mathematics. Techniques that do not generalize beyond that topic are therefore far less useful than those that do. If someone wishes to study real analysis in its own right, many universities have advanced real analysis classes, but the real purpose for undergraduate real analysis classes seems to be to set a student up for their mathematics career.
Now, you might ask why first years and second years get taught real analysis as the springboard for learning proofs, instead of say number theory, which is a very interesting question in its own right. I took real analysis my first year, but I had been previously introduced to proofs in the context of number theory. But I think it's thought to be easier to use real analysis, as calculus intuitions often carry over, and the fact that everyone already knows a lot of the computational aspect speeds up the class.
