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''. (Note added 5/18/10: De
Lillo's book is out of print, but this proof appears in many other
books. One is "Advanced Calculus", second edition, by Patrick
Fitzpatrick, and another is "Elements of Real Analysis", By David
Sprecher. Sprecher's book is published by Dover, so is not too
expensive.) 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).

Mon May 5 12:53:33 CDT 1997