Functions of Bounded Variation

Throughout this book we've encountered the theme that C(X) determines X. Said another way, to fully understand X we want to understand C(X) as well. Taking this one step further, though, raises a curious question: How are we to understand C(X) without knowing something about C(C(X))? If we want to be true to our principles, we will have to consider continuous real-valued functions on C(X). If that sounds too esoteric to bother with, fear not. As it happens, we need only to consider the continuous linear real-valued functions on C(X), and such functions have a simple and altogether user-friendly description: Definite integrals! But we're getting a little ahead of ourselves. We'll talk about integrals in the next chapter. For the present, we'll content ourselves with the study of a class of functions that turns out to be of paramount interest in this postponed discussion of integration.

To motivate the inevitable blur of definitions ahead of us, let's consider a simple example. Suppose that f(t) = (x(t), y(t)), for a ≤ t ≤ b, is a “nice” curve. What would we mean by the length of this curve?