Can One Visualize a Continuous Nowhere Differentiable Function?

Abstract

When students first encounter an example of a continuous nowhere differentiable function in a real analysis course, what do they visualize? It is not easy to visualize an infinite sum of functions when the partial sums get increasingly complicated. We offer a geometric approach via the graph of such a function. Our theme is based on the fact that, if the graph of a function intersects all non-vertical lines in a special way, then that function cannot have a derivative at any point. We will require that at each point P on the graph G of the function there must be many lines L through P having P as a limit point of the intersection . This picture is easy to imagine but impossible to represent graphically: it avoids all of the computational complications that faced nineteenth-century mathematicians when they first attempted to describe these functions.