Does $\displaystyle \left\{\begin{array}{lcl}0&:&y\leq 0\text{ or }y\geq x^2\\ \sin(\pi y/x^2)&:&0<y<x^2\end{array}\right.$ converge at $(0,0)$? [closed]

determine whether $f(x,y)$ converges in $\mathbb{R}$ as $(x,y)\to (0,0)$ in $\mathbb{R}^{2}$. Justify the answer by quoting in full every theorem or exact reference for each.

$$f(x,y) = \begin{cases} 0 & \text{if }y\leq{0}\text{ or } y\geq{x^{2}}\\ \sin(πy/x^{2}) & \text{if}\,0<y<x^{2}\end{cases}$$

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If you want somebody to do your homework, then please say so in a polite way. – Elmar Zander Mar 2 at 9:18
You just posted a similar question. Have you had a look at the answer posted to that one to see whether it helps you understand how to do this one? Or do you just want someone to do all your work for you? math.stackexchange.com/questions/318502/how-to-solve-this – Gerry Myerson Mar 2 at 9:18
sorry. i dont really clear how i can start doing them. I cant find any examples. i still have to do 6 of them. so i wanna get some help and i will do the remaining ones – Kayne Mar 2 at 9:23
and sorry again. i dont know there is a tag for homework >< – Kayne Mar 2 at 9:27
How would you approach the problem, when you have a piecewise defined single variable function? You would check what happens at the boundaries, right? You should still check that here first, but the point of the exercise is that this is not enough with two variables. So are expected to find another in-between way of approaching the origin such that the function behaves differently. Can you think of curves, along which the function would be constant? – Jyrki Lahtonen Mar 2 at 9:28
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closed as too localized by BenjaLim, Alexander Gruber♦, Davide Giraudo, Norbert, sdcvvcMar 2 at 14:06

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