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Is there any standard method to prove that an arbitrary $f:\mathbb{R}^{2}\mapsto\mathbb{R}$ function is continuous in a given $\left(x_{0},y_{0}\right)$ point?

If I consider the definition of continuity in the case of $f:\mathbb{R}\mapsto\mathbb{R}$, then I take the limits of $f(x)$ from the left and from the right side in $x_0$. If $$\lim_{x\rightarrow x_{0}-}f\left(x\right)=\lim_{x\rightarrow x_{0}+}f\left(x\right)=f\left(x_{0}\right),$$ then I say $f$ is continuous in $x_0$, but I can't do this in $\mathbb{R}^{2}$ because there are infinity possibilities to approach $\left(x_{0},y_{0}\right)$. (In the case of $f:\mathbb{R}\mapsto\mathbb{R}$ there are only two ways to approach $x_0$: from the right and from the left.)

What can I do in $\mathbb{R}^{2}$ or more generally in $\mathbb{R}^{n}$?

It is much easier to prove something is not continuous, because if I find at least one case when $$\lim_{\left(x,y\right)\rightarrow\left(x_{0},y_{0}\right)}f\left(x,y\right)\neq f\left(x_{0},y_{0}\right)$$ happens, then I can say $f$ is not continuous in $\left(x_{0},y_{0}\right)$, but it is not the question just a remark.

If there is no standard method then can you say any trick what could be useful here?

Kapes Mate
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    Techniques for proving continuity in $\Bbb{R}^2$ is a very broad topic, and needs further focus. Your preference for one-sided limits suggests to me that you're used to evaluating continuity of piecewise-defined functions, which are commonly used as practice questions to teach the concept of continuity, but are certainly not the be-all and end-all of continuity problems. Are you primarily interested in these piecewise functions in $\Bbb{R}^2$? Because narrowing your question to such functions will make the scope far more reasonable. – Theo Bendit Mar 21 '21 at 18:49
  • A technique that works with some frequency is to switch to polar coordinates centered at $(x_0, y_0)$. If your function is independent of $\theta$, this technique converts the limit to the one-dimensional case. – Robert Shore Mar 21 '21 at 20:00
  • Yes, I am interested in piecewise functions. I have never ever tried this polar coordiantes switching technique to check if something is continuous. I will have a little research about this. – Kapes Mate Mar 21 '21 at 23:19

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