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## A continuous, nowhere differentiable function.

First define a saw-tooth function f(x) to be the distance from x to the integer closest to x. Here's a plot of f:

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).

Tom Vogel
Mon May 5 12:53:33 CDT 1997