Limits of Multivariate Function Here is the problem I'm trying to solve, but having some difficulty. I'm not too sure on where to start, other than considering cases of $x$ and $m$, which determine the value of $y$ and thus the value of $f(x,y)$. I know that limits of multivariate functions hinge on approaching some point from various "paths," but I have no intuition in this case on which paths to consider. 
The problem is as follows.
"Take $f(x,y)$ to be defined as
\begin{cases} 
0, \ \text{if $y \leq 0$ or $y \geq x^4$} \\
1, \ \text{if}  \ 0 < y < x^4
\end{cases}
I want to show (a) that $f(x, y) \to 0$ as $(x,y) \to 0$ along any path of the form $y = mx^a$ for $0 < a < 4$; (b) that $f(x,y)$ is discontinuous at $(0,0)$; and (c) that $f$ is discontinuous on two entire curves."
I'd appreciate any helpful insights. I'm hoping for some help on where to start and maybe how to think of this problem intuitively, rather than the answer. 
 A: Hint: First, try to graph the curve $y=x^4$ on the plane $\mathbb R^2$. What region in the plane corresponds to the condition $y \leq 0$? (This one should be immediate: it’s the lower half-plane with its border.) What region corresponds to the condition $y \geq x^4$? (Sidenote: in my analysis courses, this was very pictorially called the epigraph of the function $x\mapsto x^4$.) In the union of these two regions your function holds the value $0$. Call it $R_0$. What is the complentary of this union of regions, i.e., where does your function hold the value $1$ instead? Call it $R_1$. What is the shape of $R_1$ near the origin?
We are concerned with what happens in neighborhoods of the origin, so focus on the unit square $[0,1]^2$ for the moment. What is the behavior of monomial curves $y = x^a$ when $0 < a< 4$ in this square? (Graph a few instances.) Do their graphs lie in the region where $f$ vanishes, $R_0$, or where it does not, $R_1$, or a little bit of both? What changes when the monomial is multiplied in front by a constant $m$? (I must guess that this $m$ may be any real number, even though the assignment did not specify.) Pick a value of $a$, fix it, and let $m$ vary in the reals. What happens when $m$ is a (small) positive number? What happens when it is $0$? What happens when $m$ is a (small) negative number? (You need to get out of the unit square for this last one, but you should see that it’s still easy to establish which region these graphs lie in.) Now take a sequence of points that all lie on the graph of your favorite monomial of this kind, in such a way that the sequence approaches the origin. What is the value of $f$ at each of these points? What may you conclude about (a)?
A suggestion for (b): to solve this, you need to remember that (1) the limit of a function as the variables approach a point in the plane exists if and only if the limits of all restrictions of $f$ to some path exist and agree with each other, and that (2) a function is continuous at a point in its domain if and only if its value at that point coincides with the limit as the variables approach that point. You have already showed in (a) that on some of the restrictions your limit exists and is $0$. What is the value of $f$ at the origin? (In which of those regions, $R_0$ or $R_1$, does the origin lie?) Are there paths (i.e. restrictions of the domain) along which the limit does not exist, or does not agree with the value $0$? You just need to propose one example of either case to be done. (Hint in a hint: it’s the second case, and you need to think monomials again!)
I’m not sure I understand (c). Do they mean discontinuous at the origin again? Or elsewhere suffices? And what is an entire curve?
