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prove a function is continuously differentiable
Use implicit function theorem to prove the following function is continuously differentiable
$f(x,y)=\begin{cases}\arctan\left(\frac{y}{x}\right) & x\neq 0 \\ π/2 & x=0,\ y>0 \\ −π/2 & x=0,\ y<0\end{cases}$ where $f$ is defined on $\mathbb{R}^2\setminus\{(0,\ 0)\}$.
How can I use implicit function theorem to prove this? I know how to directly prove it but don't really know how to prove this with IFT. any help would be really appreciated!