So my teacher was very vague when introducing this concept and most of the class didn't get it. We're analyzing the continuity of functions from $\mathbb{R^2}$ into $\mathbb{R}$
The exercise asks to analyze the continuity of the following function:
$$f(x,y) = \frac{xy+1}{y}x^2,\text{ for } y\neq 0$$ $$f(x,y) = 0 \text{ for } y=0$$
My interpretation is that the function isn't continuous, and I should be able to prove that the limit on this function does not exist along any points nearing the $y=0$ line. But I simply don't know how to do it
Another interpretation is that if I can prove there is at least one point $(x,0)$ such that the limit for $f(x,y)$ is different from $0$ near it, then I can say the function is not continuous, although this is a much weaker assertion
Thanks in advance
Choose the path $\;y=mx^2\;,\;\;m\neq0\;$ a parameter, so that the limit you want is
$$\lim_{x\to0}\frac{mx^3+1}{mx^2}x^2=\lim_{x\to0}\frac{mx^2+1}m=\frac1m$$
and thus the limit depends on $\;m\;$, so that the function's limit doesn't axist at the origin.