Lebesgue's Differentiation Theorem

In the last several chapters, we have raised questions about differentiation and about the Fundamental Theorem of Calculus that have yet to be answered. For example:

• For which f does the formula hold? If f′ is to be integrable, then at the very least we will need f′ to exist almost everywhere in [a, b]. But this alone is not enough: Recall that the Cantor function f : [0, 1] → [0, 1] satisfies f′ = 0 a.e., but.

• Stated in slightly different terms: If g is integrable, is the function differentiable? And, if so, is f′ = g in this case? For which f is it true that for some integrable g?

In our initial discussion of the Stieltjes integral, we briefly considered the problem of finding the density of a thin metal rod with a known distribution of mass. That is, we were handed an increasing function F(x) that gave the mass of that portion of the rod lying on [a, x], and we asked for its density f(x) = F′(x). We side-stepped this question entirely at the time, defining a new integral in the process, but perhaps it merits posing again.