Determine if the function that is defined as
$\displaystyle{f(x,y)= \frac{2x^2y+y^3}{x^2+y^2}}$,
for $(x,y)\ne (0,0)$, and is $0$ in origo,
is
a) continuous in origo,
b) have partial first derivatives in origo,
c) is differentiable in origo,
d) is of type $C^1$ in any environment around origo.
So for the solutions:
a) I presume that if I want to calculate the partial derivative of f(x,y) at any point away from the origin $(0,0)$, I can use the usual formulas. I would usually use $$\lim _{\left(x,y\right)\to \left(0,0\right)}f\left(x,y\right)=f\left(0,0\right)$$ However, if I want to calculate $\frac{\partial f}{\partial x}(0,0)$, I have to use the definition of the partial derivative. Is this true and how would it apply to this specific problem?
b) I think I will get the answer to this if I use the partial derivative formula in a)
c) I know that if a) is true, this one is also true
d) I don't really understand $C^1$, what is the difference from $C^2$ and how does it effect my approach to a solution?
I would really like to understand this, but also learn from a solution. I think that I am on the right way, I just have problems approaching it, or put my thinking into action.
