Function Spaces C^k: Understanding Facts

[Bachelier]

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

 

[micromass]

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?

 

[Bachelier]

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.

 

[micromass]

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.