Suppose that $f(t)$, $g(t)$ have a 2nd order derivative.
Let $F(x, y) := x f(\frac{y}{x}) + g(\frac{y}{x})$.
Then, $F_{xy}(x, y) = F_{yx}(x, y)$.
$F$ is not a class $C^2$ function in general.
Is there a necessary and sufficient condition to be $F_{xy}(x, y) = F_{yx}(x, y)$ when $F(x, y)$ has all the 2nd order partial derivatives?