Would like some guidance. What I've done so far is included.
Given,
$f(x,y)=\begin{cases} 0, \text{ if } (x,y)=(0,0)\\ \\ \frac{xy}{\sqrt{x^2+y^2}}, \text{ if } (x,y)\ne (0,0) \end{cases}$
Prove
a. $f$ is continuous (at all points)
Find a function that bounds f. Take the limit, this should show this. I can't find a bounding function.
b. $f$ has partial derivatives (at all points)
I take the $\partial_u f= \nabla f \cdot u $. I use $u = \sin x, \cos x$. I've got $f = \cos x \sin x$. but am not sure what to do it.
c. $f$ is not differentiable at $(0,0)$.
Show that the limit approaches two different points. I can't find paths such that.