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.)