Is there any notion, in the literature, of generalizing the notation $C^n$ where $n \in \mathbb{z}$, which represents smoothness (or, more precisely, existence and continuity of the $n$th derivative), to $C^x$ where $x \in \mathbb{R}^{+} - \frac{1}{2}$? For example, maybe $C^{1/2}$ would represent the functions which are at least once-differentiable but would not imply anything about its continuity, let alone further differentiability, on the input set (which I have suppressed). That seems like a natural enough specification of meaning for that particular 'index'. I know that there are different types of 'continuity' and that there is probably no complete canonical list of them, but they do form a hierarchy and we should be able to manually assign indices to them in, I hope, a bijective manner (even if there is no means by which to specify a general rule for such assignments). Actually, it is a partially-ordered hierarchy, so one particular path would have to be taken, but that is still pretty fine to me; alternatively, we could introduce another index which determines the path through the continuities taken, or something like that.
I recognize that there are separate notations (I have seen $D^n$ for the class of $n$-times-differentiable functions, for example), but that is just a matter of there existing two separate notations which sometimes coïncide.
