Let $f(x,y)=$|xy|. Note that the domain of f is all of $R^2$ and $f$ is continuous everywhere on $R^2$.
Suppose $(a,b$) lies on one of the axes, but is not the origin. That is, either $(a,b)=(a,0)$ with $a≠0$, or $(a,b)=(0,b)$ with $b≠0$.
Show that $f$ is not differentiable at $(a,b)$ by showing that one of the partial derivatives fails to exist.
and show that both partial derivatives of $f$ exist at $(0,0)$.
$f$ should also be differentiable at $(0,0)$, would I need to prove it by explicitly using the limit definition?
The first part is asking you to examine the limit defining the partial derivatives at a point of the form $(a,0)$ (or $(0,b)$, I leave this to you), and show one does not exist. This means computing $$ \frac{\partial f}{\partial y}(a,0)=\lim_{h\to 0}\frac{f(a,h)}{h}=|a|\lim_{h\to 0}\frac{|h|}{h} $$ which does not exist (why?). Note that $a\ne 0$ is important here.
At the origin, it is easy to see that the partial derivatives vanish. Then, it suffices to note that (this is the definition of the derivative, plugging in what we want $\nabla f(0,0)$ to be), $$ \lim_{h\to 0}\left|\frac{f(xh,yh)}{h} \right|=|xy|\lim_{h\to 0}\frac{h^2}{|h|} =0 $$ as required.