I am working through Ted Shifrin's book Multivariable Mathematics. There is an exercise problem that is meant to demonstrate that one can have $\dfrac{\partial^2 f}{\partial x \partial y} = 0$ but$ f(x,y) \neq g(x) + h(y)$.
The question (3.6.11) is as follows: $$ \mathrm{Given} \; f(x, y) = \begin{cases} 0, \; x < 0\; \mathrm{or} \; y < 0 \\x^3, \; x \geq 0 \; \mathrm{and} \; y > 0 \end{cases} $$
Show that $f$ is $C^2$
Show that $\dfrac{\partial^2 f}{\partial x \partial y} = 0$
Show that $f(x,y)$ cannot be written as $g(x) + h(y)$ for appropriate functions $g, h$.
I see that the domain is the entire plane except the x-axis, that is $\mathbb{R}^2 -\{y=0\}$. The function is then 0 in all quadrants except the first, where it is $x^3$.
I could show 1. and 2. above, but I am puzzled by two things.
Q1 What's wrong with writing $f(x,y) = g(x) + h(y)$ piecewise in each quadrant ?
Q2. Will anything change if the domain allows the line $y=0$ also ?
Q3. What is the takeaway from this problem ? I do not understand that.