Can Differentiability of a Function be Extended to its Absolute Value?

[omri3012]

Homework Statement

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

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...

 

[quasar987]

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.

 

[omri3012]

Thanks,

That was very helpful :)