The question looks like this. Let ##f(x, y)## = 0 if [itex]y\leq 0[/itex] or [itex]y\geq x^4[/itex], and [itex]f(x, y)[/itex] = 1 if [itex]0 < y < x^4 [/itex]. (a) show that [itex]f(x, y) \rightarrow 0[/itex] as [itex](x, y) \rightarrow (0, 0)[/itex] along any path through (0, 0) of the form [itex] y = mx^a [/itex] with [itex]a < 4[/itex]. (b) Despite part (a), show that [itex]f[/itex] is discontinuous at (0, 0) (c) Show that [itex]f[/itex] is discontinuous on two entire curves. What I've came to conclusion is that when [itex] x<0, m>0 [/itex], and [itex]a[/itex] being an odd number, [itex]y[/itex] becomes smaller then zero, so [itex]f(x, y)[/itex] can't be any larger than zero. But I don't think that's not enough. I think I need to find a way to generalize that [itex] mx^a (a<4) [/itex]is larger than [itex]x^4[/itex] or smaller than 0 when [itex]x[/itex] and [itex]y[/itex] is close enough to zero, where I cant' quite get to. In regarding (b), I know [itex]f(x, y)[/itex] is discontinuous on certain directions, but can't elaborate it in decent form. In regarding (C), How can I show it?