I believe it, but I don’t see it

Abstract

Since its discovery by Cantor in the 1870s, the one-to-one, onto function from the unit interval to the unit square has been a subject of continual interest. A corresponding function from the square to the interval, however, has rarely been considered, in spite of its unusual characteristics. This Classroom Note demonstrates how to create such a function, highlighting the counterintuitive nature of its continuities, discontinuities and slopes. This will likely be the first function whose graph students of multivariable calculus cannot visualize, and the particular results presented will surprise them. Nevertheless, guided by their instructor, using methods they have used in more standard contexts and helped by some brief hints in the text, they will be able to prove these results. While some familiarity with transfinite numbers and set concepts will enhance the experience, this is not strictly necessary. A goal is to give serious students a sense of participation in the investigation of such a function, knowing that they are not simply repeating work that others have done many times before.

Keywords:

• cantor
• continuum
• bijection
• unit interval
• unit square