# A function bounded and differentiable, but have an unbounded derivative?

1. Jul 20, 2011

### danielkyulee

Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?

2. Jul 20, 2011

### Kreizhn

Sure, so long as you properly choose your domain.

So let's think about this, we want a function whose derivative approaches infinity, but such that the function itself is bounded. Probably the easiest way to do this is to find a function whose derivative goes to zero, then rotate about the axis y=x.

You don't need anything complicated. A monomial will do fine.

3. Jul 20, 2011

### Fredrik

Staff Emeritus
Try to think of a function with a graph that looks a bit like the graph of a periodic function, but is such that the distance between the peaks get shorter and shorter instead of staying the same.

4. Jul 20, 2011

### Kreizhn

Very nice. Perhaps somewhat more complicated, but nice.

5. Jul 20, 2011

### Ray Vickson

Sqrt(x), 0 < x < 1.

6. Jul 20, 2011

### Fredrik

Staff Emeritus
Ray, here in the homework forum it's important to answer with hints, not complete solutions.