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.