(Excercise 20.3, P. Billingsley, Probability and measure) Suppose two-dimensional distribution function $F$ has continuous density $f$. Show that $$f(x,y)=\frac{\partial^2F(x,y)}{\partial x \partial y}.$$
This is incorrect! For example, the density $$ f(x,y)= \frac{1}{2\sqrt{2\pi}} |x| \exp\left(-|x|-\frac{1}{2}x̣^2y^2\right)$$ it does not verify the affirmation.
Under what conditions this relation between $F$ and $f$ is valid?