Differentiability of a Multivariable function

I’m having a little confusion about part b of this question as to why I am allowed to use the limit definition of a partial derivative.
Here’s what I think:
I know that y^3/(x^2+y^2) is undefined at the origin but it does approach 0 when it GETS CLOSE to the origin. So technically defining f(x,y)=0 fills in that hole and it becomes a smooth curve and so I can use the limit definition? (Because the geometric interpretation of a partial derivative, at least with respect to x, is the intersection of y=y_0 with the surface, which becomes a 2d curve, and then I take the derivative wrt x.)
If instead f is defined to be some other number like 2 at the origin then this will not work?

The partial derivative at ##(0,0)## is defined as:
$$\frac{\partial f}{\partial x} = \lim_{h \to 0}\frac{f(h, 0) - f(0, 0)}{h}$$PS I thought partial derivatives were only defined for a continuous function, but apparently not!