Suppose $f:(x,y)\in K\times \mathbb{R}\to \mathbb{R}$ be a continuous function (in $x$ and $y$) and $K$ is a compact set. If $f$ is continuously differentiable in $y$. Is $f_y$ still continuous in $x$?
I am not sure whether it is correct or under what condition it is correct. Briefly speaking, if $f$ is continuously differentiable in $y$, then for any $x\in K$, I have $$ f(x,y+h)=f(x,y)+f_y(x,y)h+o(h), $$ as $h\to0$. I would like $f_y(x,y)$ to be bounded for all $x\in K$ and $|y|\le C$, and a sufficient condition is to show $f_y$ is still continuous in $x$.
Try $f(x,y) = \begin{cases} \sqrt{|x|} \arctan {y \over x},& x \neq 0 \\ 0, & \text{otherwise}\end{cases}$.
Then ${\partial f(x,y) \over \partial y} = \begin{cases} {\sqrt{|x|} \over x^2+y^2},& x \neq 0 \\ 0, & \text{otherwise}\end{cases}$.
Then, for a fixed $x$, the function $y \mapsto {\partial f(x,y) \over \partial y}$ is continuous, but $x \mapsto {\partial f(x,0) \over \partial y}$ is not continuous at $x=0$.