Function Spaces C^k: Understanding Facts

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Discussion Overview

The discussion revolves around the understanding of function spaces, specifically the notation and implications of differentiability classes, particularly ##C^k## and ##C^{-1}##. Participants explore definitions, differences between terms, and the characteristics of functions within these spaces.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question whether "function space" and "Differentiation Class" are equivalent terms, with one suggesting that function spaces are more general.
  • It is proposed that a function ##f## belongs to ##C^k## if its k-th derivative exists and is continuous, which some participants agree with.
  • There is uncertainty regarding the classification of functions with discontinuities as belonging to ##C^{-1}##, with one participant expressing doubt about the standard use of this terminology and seeking references.
  • Another participant mentions that ##C^{-1}## may refer to functions that are piecewise continuous, indicating a limited number of discontinuities, but lacks a definitive reference.

Areas of Agreement / Disagreement

Participants generally agree on the definition of ##C^k## but express disagreement or uncertainty regarding the classification of ##C^{-1## and its standard terminology.

Contextual Notes

There is a lack of consensus on the definition and implications of ##C^{-1}##, with references and standard terminology remaining unresolved.

Bachelier
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Could you check my understanding of these facts please:

First is there a difference between the notations "function space" and "Differentiation Class" ? Methinks they are one.

Second, a function ##f## is said to be in ##C^k \iff \ f^{(k)}## exists and is in ##C^0##.

Third, if a function has discontinuities then it belong to ##C^{-1}##

Thank you
 
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Bachelier said:
Could you check my understanding of these facts please:

First is there a difference between the notations "function space" and "Differentiation Class" ? Methinks they are one.

A function space is something more general. The spaces ##C^k##are certainly function spaces, but there are many more.

Second, a function ##f## is said to be in ##C^k \iff \ f^{(k)}## exists and is in ##C^0##.

Yes.

Third, if a function has discontinuities then it belong to ##C^{-1}##

I have encountered ##C^{-1}## before, but I forgot what it was. I kind of doubt that it is standard terminology anyway. Do you have a reference for this?
 
micromass said:
I have encountered ##C^{-1}## before, but I forgot what it was. I kind of doubt that it is standard terminology anyway. Do you have a reference for this?

Unfortunately I can only quote the "Wikipedia" page on "Smooth Functions". I tried to do a google search for the term to no avail.
 
Bachelier said:
Unfortunately I can only quote the "Wikipedia" page on "Smooth Functions". I tried to do a google search for the term to no avail.

The way I've seen it defined is that ##C^{-1}## has functions that are piecewise continuous. So they have discontinuities, but only a limited number. But I don't have a reference either. It's not very important anyway.
 

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