I've just learned the implicit function theorem from my course, and I found that I have some trouble when doing the exercises, that I don't know how to actually apply the implicit function theorem.
Here is a simple example from my exercise sheet:
$\textbf(Exercise)$
Show that the equation $$xy^4 - x^3 +y = 0$$ defines an implicit function $\phi:x \to y$ in a neighbourhood of $0$ and that $\phi(0) = 0$.
My question is, first of all, I don't know where should I start the proof, and then what does neighborhood here mean?
I understand my question could be stupid, I searched for some similar questions on the Internet but the difference made me more confused.
Thank you very much in advance.