A function that is surjective on every interval

by David Radcliffe

The intermediate value theorem states that if f is a continuous real-valued function on the closed interval [a, b], and if c is any real number between f(a) and f(b), then there exists c in [a, b] such that f(x) = c. A function that satisfies the conclusion of this theorem is called a Darboux function.

Although every continuous function is a Darboux function, it is not true that every Darboux function is continuous. Perhaps the simplest example is f(x) = sin(1/x) for x not 0, f(0) = 0. The graph of this function is known as the topologist’s sine curve. The importance of this curve lies in the fact that it is connected but not path-connected.

This function is only discontinuous at 0, but the British mathematician John H. Conway constructed a Darboux function that is discontinuous at every point. In fact, it has the stronger property that it is surjective on every nonempty open interval. That is, if a, b, and y are real numbers with a < b, then there exists a real number x such that f(x) = y. This function is called Conway’s base 13 function, because it is defined in terms of the base 13 digits of the argument.

I wish to propose another example of a function that is surjective on every interval. The function is defined as follows:

f(x) = \displaystyle \lim_{n\to\infty} \tan(n!\, \pi x) if the limit exists,

f(x) = 0 otherwise.

In addition to being surjective on every interval, my function has a number of other appealing properties. It is defined using a simple formula instead of arcane digit manipulations. The function is equal to zero almost everywhere. Most remarkably, it is periodic and every positive rational number is a period.

I challenge the reader to verify these claims for herself or himself. My proof is available here.