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 $\displaystyle C^1$ 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 $\displaystyle C^2$ 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:

$\displaystyle g (x) = \frac{f (x)}{(x)}$ if $\displaystyle x \neq 0 $

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

$\displaystyle f (x) = f (0) + xf '(0) + x ^ 2 \frac{f"(0)}{2} + o (x ^ 2) $

or f (0) = 0 therefore

$\displaystyle f (x) = xf '(0) + x ^ 2 \frac{f"(0)}{2} $

and

$\displaystyle \ frac{f(x)}{x} = f '(0) + x \frac{f"(0)}{2} = g (x) $

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)