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 is differentiable on and

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

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