there is a problem in the calculus book I'm reading that has a function f (see below) that is not continuous at (0,0). the book says that despite this fact, both partial derivatives still exist there. so i am trying to use wmaxima to prove the existence of just fx for now. So i go through the motions and wmaxima gives me an answer that appears to be validate, but I don't understand the answer. so I'm not convinced. I think I'm missing some key point either in how maxima takes derivatives and evaluates them, or in my understanding of partial derivatives. I'm not sure which.
edit: oops. i try to get a better quality picture for you.
edit: here is the actual problem I'm trying to solve.
I cannot explain the xMaxima behaviour (perhaps having produced an error with parallel substitution, it then tries serial substitution to try to get an error-free answer).
The original point about the partial derivative is that $g(x)=f(x,0)=0$ for all $x$ and so is continuous including at $x=0$, and has a partial derivative with respect to $x$ of $0$. So $f(x,y)$ has a partial derivative everywhere of $\dfrac{\partial f}{\partial x} = \dfrac{y(y^2-x^2)}{x^2+y^2}$ except when $y=0$ where it is $\dfrac{\partial f}{\partial x} = 0 $.
You can make a similar statement about $h(y)=f(0,y)=0$ and so $\dfrac{\partial f}{\partial y}$.