Suppose that $D$ is an area, $n$ is a natural number. Show that: $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}$ is continuous on $D$.
It's easy to prove necessity, but I can't prove sufficiency.
My idea is so direct, and I want to find that the expression of $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}$, then I can find an expression which is indirectly related to $\frac{\partial^n f}{\partial x^n}$ and $\frac{\partial^n f}{\partial y^n}$. So I try to use partial differential operators, we can prove that $$\frac{\partial^n }{\partial z^k\partial\bar{z}^{n-k}}=\frac{1}{2^n}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)^k\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)^{n-k}$$ $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are not interchangeable. But I found that the equation above seems to have no practical use, so my idea may be wrong.
I've thought of it, just use induction and notice a formula like this:$$ \frac{\partial^n f}{\partial \bar{z}^n}=\frac{\partial^{n-1} }{\partial \bar{z}^{n-1}}(\frac{\partial f}{\partial \bar{z}}); \frac{\partial^n f}{\partial z\partial\bar{z}^{n-1}}=\frac{\partial^{n-1} }{\partial z \partial\bar{z}^{n-2}}(\frac{\partial f}{\partial \bar{z}}); \frac{\partial^n f}{\partial z^2\partial\bar{z}^{n-2}}=\frac{\partial^{n-1} }{\partial z^2\partial\bar{z}^{n-3}}(\frac{\partial f}{\partial \bar{z}}); \cdots; \frac{\partial^n f}{\partial z^n}=\frac{\partial^{n-1} }{\partial z^{n-1}}(\frac{\partial f}{\partial z}) $$