# Fundamental Theorem of Calculus properties

1. Apr 2, 2009

### am100181

1. The problem statement, all variables and given/known data

Find a function f : [-1,1] ---> R such that f satisfies the following properties:

a) f is continuous
b) f is restricted to (-1,1) is differentiable
c) its derivative f' is not differentiable on (-1,1)

2. Relevant equations

3. The attempt at a solution
I kinda think that the mean value theorem and Theorem 2 of the fundamentals $$\int$$f(x)dx = F(b)-F(a) got some link but I can't seem to get it. I do understand that for f'' not to exist, x should be undefined on the (-1,1). Please help.

2. Apr 2, 2009

### am100181

A read somewhere that a hint would be to begin with an absolute value and use $$\int$$f(x)dx = F(b)-F(a) (fundamental theorem of calc prep 2) repeatedly.. but still puzzled

3. Apr 2, 2009

### Dick

Start with c). Pick a nondifferentiable function on (-1,1) and integrate it to get f.

4. Apr 3, 2009

### HallsofIvy

Staff Emeritus
|x| is a very simple function that is not differentiable at x= 0.

5. Apr 3, 2009

### sutupidmath

You are right, but the OP is looking for a function such that it is once differentiable on (-1,1) but not twice, and is continuous of course on the same interval.

Edit: ignore it!

6. Apr 3, 2009

### HallsofIvy

Staff Emeritus
Yes, and combining |x| with Dick's suggestion gives exactly that!

(Edit: Too late! I gotcha!)