• Support PF! Buy your school textbooks, materials and every day products Here!

Definition of differentiation

  • Thread starter omri3012
  • Start date
  • #1
62
0

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...
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8


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
62
0


Thanks,

That was very helpful :)
 
Last edited:

Related Threads for: Definition of differentiation

Replies
2
Views
2K
Replies
1
Views
7K
Replies
3
Views
3K
Replies
3
Views
1K
Replies
4
Views
1K
Replies
4
Views
2K
Replies
2
Views
1K
Top