# Definition of differentiation

1. May 23, 2009

### omri3012

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

By the formal definition of differentiation Prove that if f differentiable in c and f(c)$$\neq0$$ then |f| differentiable in c.

3. The attempt at a solution

I know that if f differentiable do it also continues but I stuck because this fact correct necessarily only for one direction...

2. May 23, 2009

### quasar987

Re: diffrentiability

Let c be such that f(c) is not 0. Recall that since f is differentiable, it is continuous.

Suppose first that f(c) >0. Intuitively, the fact that f is continuous implies that as you wobble a little around c, f wobbles a little around f(c), and so for small enough wobbling around c, f should remain of the same sign as f(c). In other words, if our understanding of continuity is correct, we should be able to prove that there exists an $\epsilon>0$ such that f(x)>0 for all x in $(c-\epsilon,c+\epsilon)$. In particular, this means that |f(c+h)|=f(c+h) for small enough h.

And in the same way, we can show that if f(c)<0, then |f(c+h)|=-f(c+h) for small enough h.

Hopefully, you see how you can use this to calculate the derivative of |f| at c.

3. May 23, 2009

### omri3012

Re: diffrentiability

Thanks,

That was very helpful :)

Last edited: May 23, 2009
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