Let $\cal F^k$ be a set of functions, each of class $C^k$,
i.e., both, for every function in $\cal F^k$: 


* $k^{\textrm{th}}$ derivatives exist, and
* are continuous.


Let $D(\cal F^k)$ be the set of all those functions' derivatives,
up to the $k^\textrm{th}$ derivative.


My basic question is:



> 
> ***Q0***. What is the relation between $\cal F^k$ and $D(\cal F^k)$ ?
> 
> 
> 


If $\cal F^k$ is the set of all univariant polynomials of at most degree $k$,
then $D(\cal F^k)$ is the polynomials of degree at most $k{-}1$.



> 
> ***Q1***. Is there, in general, 
>  a nice structural relationship between $\cal F$ and $D( \cal F)$ ?
> 
> 
> 


What I am really seeking to answer is this (from the title):



> 
> ***Q2***. Does differentiation widen, or narrow—or neither!—the class of functions $\cal{F}$?
> 
> 
> 


Perhaps $D( \cal F) \subseteq \cal{F}$ ?


(One could replace "differentiation" with "integration" and ask
analogous questions.)



Does differentiation widen, or narrow, the class of functions?
The answer depends on the class you consider. If $F^k=C^k$, the class of all
functions with your properties, then $D(F^k)=C^{k-1}$,
and $C^k$ is a proper subset of $D(F^k)$. If $F^k$ is the class of all polynomials, then $D(F^k)=F^k$. If $F^k$ is the class of polynomials of degree at most $d$, then $D(F^k)$ is a proper subset of $F^k$. 


