Thread: continuity and differentiability

Hello I would just like you to tell me what would be the best method to demonstrate that the function g is of class on the interval J(I was spoken to apply Taylor Lagrange or Taylor-Young but I did not understand much) I thank:
Here is the statement:

Let f be a function of class over an interval J, such that f (0) = 0.
It is assumed that 0 is an interior point to J. We define the function g on J by:
if
and g (0) = f '(0).

What I have done yet below:

In a first time I apply Taylor Young to see where that leads me

-The formula for Taylor Young in 0 to order 2 of f gives us



or f (0) = 0 therefore

and


I wonder now if this is fair and whether it will indeed serve me something.
(I just want to say that I'm French and I think you'll find my language can be a little too basic, I hope that this will not prevent you from answering me)

The function g is clearly differentiable at all points of J other than 0. In fact, by the usual quotient rule for differentiation.

To see what happens at x=0, you have to find the derivative from first principles: . Using the definitions of g(x) when x≠0 and when x=0, you see that this is equal to . Thus g is differentiable at 0 (with derivative ).

Finally, to show that g is of class , you must check that the derivative g' is continuous at 0. To do this, evaluate (using l'Hôpital's rule) and verify that it is equal to .