Smoothness of Product Functions

Question

Suppose $f:\mathbb{R}^{n}\times\mathbb{R}^k\to\mathbb{R}$. Define $$f(x,y)\equiv f_x(y)\equiv f_y(x)$$ If $f_x$ and $f_y$ are both smooth, is $f$ smooth?

Answer 1

Well-known counterexample: $f=\frac{x^2-y^2}{x^2+y^2}$