Function differentiable, but derivative not bounded

[JG89]

Homework Statement

Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1]

Homework Equations

The Attempt at a Solution

Ok, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. I also know that it cannot be increasing with jump discontinuities, because the derivative has to have the intermediate value property. I also do not think (but am not 100 percent sure) that it can have any infinite discontinuities, because I think that would make f unbounded, which it cannot be on [0,1] if its differentiable on [0,1].

Any ideas?

 

[hgfalling]

Let's gather the facts you know about the two functions:

1) f is differentiable
1a) f is continuous (follows from differentiability)
1b) f is bounded on [0,1] (follows from continuity on a compact set)

2) f' is unbounded on [0,1]

So f' must go off to infinity (or negative infinity) on [0,1]. Yet f is bounded. So what could be happening? Hint: It seems f must be oscillating really fast. What kinds of functions do this?