Definition of differentiation

In summary, the conversation discusses how to prove that if f is differentiable at c and f(c) is not equal to 0, then |f| is also differentiable at c. The proof involves using the fact that a function is continuous if it is differentiable, and showing that for small enough values around c, f will remain of the same sign as f(c). This leads to the conclusion that |f(c+h)|=f(c+h) or |f(c+h)|=-f(c+h) depending on whether f(c) is positive or negative. This can then be used to calculate the derivative of |f| at c.
  • #1

Homework Statement

By the formal definition of differentiation Prove that if f differentiable in c and f(c)[tex]\neq0[/tex] 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...
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  • #2

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 [itex]\epsilon>0[/itex] such that f(x)>0 for all x in [itex](c-\epsilon,c+\epsilon)[/itex]. 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


That was very helpful :)
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