What it means to be infinitely differentiable

In summary, the question is whether saying "function f is infinitely many times continuously differentiable" is the same as saying "function f is infinitely many times differentiable" and which one defines the property C^infinity. The answer is that while both refer to a function being infinitely differentiable, the former specifies that the derivatives must also be continuous, while the latter only requires that the derivatives exist. It's important to note that differentiating a function "infinitely many times" is not a well-defined operation and an infinitely differentiable function is one whose Nth derivative exists for any finite integer N greater than 0. Additionally, a differentiable function is also continuous.
  • #1
fred_91
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<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

 
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  • #2
Infinitely differential [tex]C^{\infty}[/tex] 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.
 
  • #3
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.
 
  • #4
A differentiable function is continuous.
 

1. What does it mean for a function to be infinitely differentiable?

Being infinitely differentiable means that a function can be differentiated an infinite number of times. This implies that the function is smooth and has no abrupt changes in its slope or curvature.

2. How is infinite differentiability different from finite differentiability?

Finite differentiability means that a function can only be differentiated a certain number of times before reaching a point where it is no longer differentiable. Infinite differentiability, on the other hand, means that the function can be differentiated an infinite number of times without ever reaching a point where it is not differentiable.

3. What is the significance of a function being infinitely differentiable?

A function being infinitely differentiable is significant because it indicates that the function is very smooth and has a well-defined shape. This can be useful in many applications, such as in physics and engineering, where having a smooth and continuous function is important.

4. Can all functions be infinitely differentiable?

No, not all functions can be infinitely differentiable. Functions that have abrupt changes in their slope or curvature, such as a discontinuity or sharp corner, are not infinitely differentiable.

5. How can we determine if a function is infinitely differentiable?

A function can be determined to be infinitely differentiable if all of its derivatives exist and are continuous. This means that the function must have a well-defined slope and curvature at every point on its graph.

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