Now define 
to be 
. This has 
as many ``teeth'' as
f per unit interval, but their height is 
times the height of the
teeth of f. Here's a plot of 
, for example:
Finally, define h(x) to be the sum 
. For every x
this sum converges by comparison with a geometric series. It's already beyond
elementary calculus to show that h(x) is continuous (to advanced calculus
students: h(x) is the sum of a uniformly converging series of continuous
functions, hence continuous). For the proof that h(x) is not differentiable,
see appendix A of N. J. De Lillo's ``Advanced Calculus with Applications''. The
rough idea is that at every step we add more and more corners. Here's a plot of
h(x): (actually only a partial sum rather than the infinite sum).
