Suppose that $D$ is a connected open set, $n$ is a natural number. Show that: $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}(0\leq k\leq n)$ is continuous on $D$.
$\quad$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 $$\frac{\partial^n }{\partial z^k\partial\bar{z}^{n-k}}=\frac{1}{2^n}(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y})^k(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y})^{n-k}$$ $\frac{\partial}{\partial x}$ 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.