angels’ staircases

Here’s a pair of facts I hadn’t much considered before the last time I taught real analysis:

The only kind of discontinuity a monotone function can have are jump discontinuities.
A monotone function can have at most countably many discontinuities.

One thinks immediately of the floor function \(f(x) = \lfloor x \rfloor\), which has a jump of size 1 at every integer. It is possible to have infinitely many jumps within a finite interval; for instance, \(\frac{1}{\lfloor1/x\rfloor}\) has a jump at every unit fraction \(1/n\). But can a monotone function have a dense set of discontinuities, say, the set of rationals? Sure, here’s one:

That’s the graph of \[f(x) = \sum_{n=1}^\infty \frac{\lfloor nx\rfloor}{2^n}.\] It has a jump of size \(1/(2^q-1)\) at each fraction of the form \(p/q\) (when \(p/q\) is in reduced form, of course). Exercise. Why are these the sizes of the jumps? Keep in mind where the discontinuities of \(\lfloor nx \rfloor\) appear.

Here’s a general way to construct a monotone function \(f : \mathbb{R}\to\mathbb{R}\) whose set of discontinuities is your favorite countable set \(C\). Let \(c_1\), \(c_2\), \(c_3\), \(\dots\) be an enumeration of \(C\), and for each \(x\in\mathbb{R}\), set \[N(x) = \{ n\in\mathbb{N} : c_n \le x \} \] Let \(a_1\), \(a_2\), \(a_3\), \(\dots\) be a sequence of strictly positive numbers such that \(\sum a_n \lt \infty\). Then define \[ f(x) = \sum_{n\in N(x)} a_n. \] This function is discontinuous at each point in \(C\) and continuous at each point that is not in \(C\). (Exercise. Why? Keep in mind that the tail of a convergent series can be made arbitrarily small.) Essentially, we have created a distribution by placing a “delta mass” of weight \(a_n\) at the point \(c_n\), and \(f\) is the integral of this distribution from \(-\infty\) to \(x\).

When \(C\) is dense, the construction above produces a strictly increasing function \(f\), and moreover the image of \(f\) is nowhere dense. I call the graph of such a function an “angels’ staircase”, because \(f\) is a one-sided inverse of a “devil’s staircase” function \(g\)—that is, \(g\) is a monotone increasing function that is constant except on a set of measure zero. (MathWorld, on the other hand, uses “devil’s staircase” to refer to both kinds of functions.)