A function $f$ is said to be of differentiability class $C^k$ if its first $k$ derivatives are continuous. It has the property that $m>n \implies C^m \subset C^n$ and that all $C^k$s are algebras.
My question is this:
Seeing as many continuous functions are the integrals of discontinuous functions, can we define $C^k; k<0$ to be the set of functions whose $k^{th}$ integral is continuous? As the integral of a continuous function is continuous, $C^0 \subset C^k;k<0$ and it’s pretty easy to check that this is a vector space, but is it an algebra?
Another question is if there exists a $C^{-\infty}$ as the intersection of all $C^k;k<0$ analogously to how $C^{\infty}$ is defined. If this set exists, is it still an algebra and what would it look like? Seeing as $C^k$ gets broader as $k$ gets smaller, $C^{-\infty}$ is actually just the limit as $k$ approaches $-\infty$.
Another thing to note would be what functions aren’t in $C^{-\infty}$ and the other $C^k$s.