What it means to be infinitely differentiable

[fred_91]

<Moderator's note: Moved from a homework forum.>

1. Homework Statement
I am wondering if it is the same to say :

Function f is infinitely many times continuously differentiable
AND
Function f is infinitely many times differentiable

And if it is not the same, which one defines: C^infinity ?

Homework Equations

The Attempt at a Solution

 

[mathman]

Infinitely differential $$C^{\infty}$$ refers to a property at a particular point. However to get the nth derivative at that point, the (n-1)th derivative needs to be continuous at that point.

 

[Stephen Tashi]

It's worth mentioning that to differentiate a function "infinitely many times" is an undefined operation. An "Infinitely differentiable" function is one whose Nth derivative exits for any finite integer N > 0.

 

[lavinia]

A differentiable function is continuous.