Bounded variation: Why do we need continuity?

Question

I use the definition of a bounded variation function that can be found here under "formal definition" (definition $1.1$ and $1.2$). My question is, do we need that $f$ is continuous in this definition or could we generalize it also for non-continuous $f$? I don't see where continuity is needed. I have also seen a similar definition for right continuous functions, but then again I don't see why that would be needed.

Answer 1

It is not strictly necessary, no. For instance, the indicator function $1_{[0, 1]}$ is discontinuous, but clearly of bounded variation on any subset $[a, b]$.

However, it can be shown that a function is of bounded variation if and only if it can be written as the difference of two non-negative and increasing functions. Such functions are continuous on all but a countable set of points, and hence the same is true of any function of bounded variation.

One reason one might aditionally require (right) continuity is that this is very useful when constructing signed Lebesgue-Stieltjes measures.