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.

 

[blue_raver22]

for your question. I am happy to check your understanding of these facts about function spaces C^k.

1. The terms "function space" and "differentiation class" are often used interchangeably in mathematics. Both refer to a collection of functions that share certain properties, such as being continuous or differentiable. So, yes, they can be considered as the same concept.

2. Your understanding of the second fact is correct. A function f is said to be in the function space C^k if and only if its kth derivative exists and is continuous (in other words, in the function space C^0). This means that the function is differentiable k times and all of its derivatives up to the kth order are continuous.

3. Your understanding of the third fact is also correct. If a function has discontinuities, it belongs to the function space C^{-1}. This means that the function is not differentiable, but its antiderivative (primitive) is continuous. In other words, the function has a discontinuity in its derivative, but its integral is still continuous.

I hope this helps to clarify your understanding of these facts about function spaces C^k. Keep up the good work in your studies!