MHB Continuity of piecewise function of two variables

[A330NEO]

The question looks like this.
Let $f(x, y)$ = 0 if $y\leq 0$ or $y\geq x^4$, and $f(x, y)$ = 1 if $0 < y < x^4 $.

(a) Show that $f$ is discontinuous at (0, 0)

(b) Show that $f$ is discontinuous on two entire curves.
In regarding (a), I know $f(x, y)$ is discontinuous on certain directions, but can't elaborate it in decent form.

In regarding (b), How can I show it?

 

[Evgeny.Makarov]

Let $f(x, y)$ = 0 if $y\leq 0$ or $y\geq x^4$, and $f(x, y)$ = 1 if $0 < y < x^4 $.

(a) Show that $f$ is discontinuous at (0, 0)

By definition, $f$ is continuous at $(0,0)$ if for every $\varepsilon>0$ there exists a $\delta>0$ such that if $(x,y)$ is located within distance $\delta$ from $(0,0)$, then $|f(x,y)-f(0,0)|=|f(x,y)|<\varepsilon$. Choose any $0<\varepsilon<1$. Can you find a circle around $(0,0)$ such that within that circle $f$ takes values $<\varepsilon$?

(b) Show that $f$ is discontinuous on two entire curves.

By the two curves, does the problem mean $y(x)=0$ and $y(x)=x^4$? Again, within every circle whose center lies on these curves, however small the radius is, there are points where $f$ returns 0 and other points where $f$ returns 1.