Proof of Smoothness: Analytical Steps & Examples

[nughret]

My lecturer gave me a question that included giving a proof that a particular function is smooth. I have taken a course on analysis and have no problems when it comes to proof of continuity; i was just wondering what the usual steps are in proving that a function is smooth.
I would guess that it would involve some sort of induction on the derivatives but if someone could sketch out a general proof that would be grand.

 

[CompuChip]

There are some rules like: sums, products and compositions of smooth functions are smooth. Usually one uses these to show that a function is smooth. Otherwise, induction on the derivatives might work.

Since it is only a small portion of your entire exercise, could you post the function?

 

[nughret]

Yes well this specific question is stated as such :
f(x) is a smooth function, prove the function

G(x) = f'(0) , x = 0
(f(x) - f(0))/x , otherwise

is smooth.
I previously assumed by product rule it is true that G(x) is smooth when x is not equal to zero but obviously the whole point of the question is about x=0.