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I just realized that I don't really know what the definition of $C^{1,2}$ (or $C^{m,n}$) means. Two candidates come to mind:

1) For every $y$, the function $x\mapsto f(x,y)$ is $C^1$, and for every $x$, the function $y \mapsto f(x,y)$ is $C^2$

2) There exist $f, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial^2 f}{\partial y^2}$ that are jointly continuous.

Is there any consensus on what the usual definition is? I have books that use this notation and never define it.

Important note: $C^{1,2}$ is common notation in stochastic calculus, it refers to different derivatives, and has nothing to do with Holder continuity for functions of one variable.

nullUser
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  • Possible duplicate of https://math.stackexchange.com/questions/1532589/regarding-notation-of-functions. – lhf Feb 04 '16 at 00:23
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    This is a bit weird. The usual thing that I see is $C^{k,\alpha}$, where $k$ is a nonnegative integer and $\alpha \in [0,1)$. When $\alpha=0$ this is just the ordinary space of functions with $k$ continuous derivatives; when $\alpha>0$ this is a Holder space, which contain functions with $k$ continuous derivatives and where the $k$th derivative is $\alpha$-Holder continuous. But I have never seen something like $C^{1,2}$ where the second number is larger than $1$. Can you give some context? – Ian Feb 04 '16 at 00:23
  • Which books?... – lhf Feb 04 '16 at 00:24
  • @lhf, not a duplicate of that question. My question is about how that question's answers are wishy-washy, not firm definitions. – nullUser Feb 04 '16 at 00:36
  • In that case it just means $C^1$ in the first argument (usually $t$) and $C^2$ in the second (usually $x$). Which is what you said in the OP (which are equivalent). – Ian Feb 04 '16 at 02:13
  • @Ian Please define what it means to be $C^1$ in the first argument and $C^2$ in the second. That is what my question is asking. What does that mean? I give two possible interpretations in my question. Does it mean either of those? Or something else? – nullUser Feb 04 '16 at 03:33
  • It is the first one; usually the second one holds as well. – Ian Feb 04 '16 at 03:54

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