Reasoning for the function $f(x,y)=x^{2}e^{y}$ is differentiable on $\mathbb{R}^{2}$.
Explanation :
the correct reason should be $f_{x}=2xe^{y}$ and $f_{y}=x^{2}e^{y}$ are both continuous on $\mathbb{R}^{2}$ , then $f$ is differentiable at every point on $\mathbb{R}^{2}$.
But my friend told me :
$f_{x}$ and $f_{y}$ both exist at each point $(x,y)\in\mathbb{R}^{2}$ and for all point $(x,y)\in\mathbb{R}^{2}$, we have $$f_{xy}=2xe^{y}=f_{yx}$$
Hence , $f$ is differentiable on $\mathbb{R}^{2}$.
I confused . Is this reason valid for testing function differentiable ? Or is there a counterexample for this ?