I am given that X and Y are independent iff $F(x, y) = F_X(x)F_Y(y)$ and are asked to show joint probability $f(x,y)=f_X(x)f_Y(y)$
So I think this is an easy proof but I am not sure If it is correct, I did this
$f(x,y)=\frac{\partial ^2}{\partial x\partial y}(F_X(x)F_Y(y))$
$=\frac{\partial}{\partial x}(F_X(x)f_Y(y))$
$=f_X(x)f_Y(y)$
I think I have seen it some where but I am not sure if $\frac{\partial}{\partial x}F_X(x)=f_X(x),\frac{\partial}{\partial y}F_Y(y)=f_Y(y)$ is true.