How rough can the graph of a continuous function be?

We know that if a function f(x) has a derivative at x=c, then f(x) is continuous at x=c. But, the converse of this theorem is false. A function need not have a derivative at a point, where it is continuous, as in the following example:

The function y=|x| is differentiable on (-\infty, 0) and

(0,\infty) but has no derivative at x=0.

Also, using this simple idea, we can use a sawtooth graph to define a continuous function that fails to have a derivative at infinitely many points.

But, can a continuous function fail to have a derivative at every point?

The answer, surprisingly enough, is yes, as Karl Weierstrass (1815-1897) found in 1872. One of his formulae (there are many others like it) was

f(x)=\sum_{n=0}^{\infty} (2/3)^{n} \cos (9^{n}\pi x)

a formula that expresses the function f as an infinite sum of cosines with increasingly higher frequencies. By adding wiggles to wiggles infinitely many times, so to speak, the formula produces a graph that is too bumpy in the limit to have a tangent anywhere.

Continuous curves that fail to have a tangent anywhere play a useful role in chaos theory, in part because there is no way to assign a finite length to such a curve. We will see what length has to do with derivatives later in one of the blog articles.

More later…

Nalin Pithwa

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