Well, here's my problem. I've been at it all night and have made zero progress:

Consider the real function

$\displaystyle f(x,y) = xy(x^2 + y^2)^{-N}$, in the respective cases N = 2, 1, and $\displaystyle \frac{1}{2}$.

a.) Show that in each case the function is differentiable ($\displaystyle \mathcal{C}^{\omega}$) with respect to x, for any fixed y-value (and that the same holds with the roles of x and y reversed).

Nevertheless, f is not smooth as a function of the pair (x,y).

b.) Show this in the case N = 2 by demonstrating that the function is not even bounded in the neighborhood of the origin (0,0).

c.) Show this in the case N = 1 by demonstrating that the function though bounded is not actually continuous as a function of (x,y).

d.) Show this in the case N = $\displaystyle \frac{1}{2}$ by showing that though the function is now continuous, it is not smooth along the line x = y.

Thanks for any and all help. It's much appreciated.