What it means to be infinitely differentiable

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Main Question or Discussion Point

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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

 

Answers and Replies

  • #2
mathman
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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
Stephen Tashi
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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
lavinia
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A differentiable function is continuous.
 

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