0
$\begingroup$

Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "differentiable, but not necessarily continuously differentiable" is rather inefficient to say the least. Surely, I can't be the first one in the history of mathematics to be faced with the same problem!

It is quite standard to denote by $C(X,Y)$ or $C^0(X,Y)$ the space of continuous maps and by $C^k(X,Y)$, $k \in \mathbb{N} \cup \{\infty,\omega\}$, the space of $k$-times continuously differentiable (resp. smooth or analytic) maps.

But is there somewhat accepted / standard notation for the set between $C^0$ and $C^1$ of differentiable, but not necessarily continuously so maps?

I guess, some possibilities are $C^{>0}(X,Y)$, $C^{0^+}(X,Y)$, or $C^{1^-}(X,Y)$, but it feels like I'm just making up stuff at this point (and chances are these notations already mean some other kinds of regularity stronger than $C^0$, but weaker than $C^1$).

Improve this question
$\endgroup$
8
  • 2
    $\begingroup$ I don't think I have seen a standard notation for these spaces. Note that especially the script +/- notation is in use in some sources for lipschitz maps $\endgroup$
    – Alexander Schmeding
    Commented Oct 12 at 10:17
  • 1
    $\begingroup$ I have never seen any notation for this set either, it is a strange, pathologically behaved set of functions. $\endgroup$
    – Nate River
    Commented Oct 12 at 10:23
  • 1
    $\begingroup$ @NateRiver: You are right, the general pathology of such functions is likely the reason why there isn't a well established notation for this set. Nevertheless, in the presence of additional structure assuming such regularity can actually make sense, e.g. Looman–Menchoff theorem. $\endgroup$
    – M.G.
    Commented Oct 12 at 10:32
  • 1
    $\begingroup$ Since continuous functions use $C$, perhaps differentiable functions should use $D$, or the same character in some other font? While it wouldn't be an accepted notation it would be a small variation on a standard for notation. $\endgroup$
    – Ryan Budney
    Commented Oct 12 at 17:44
  • 1
    $\begingroup$ I suppose you could go to other languages as well. I'm assuming you are talking about Frechet-differentiable functions, so you could use $F$ in a stylized font, as well. $\endgroup$
    – Ryan Budney
    Commented Oct 12 at 18:24

0

Reset to default

You must log in to answer this question.