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I am wondering how to check whether a multidimensional function is continuous. I.e. I am thinking of functions like

$f_1(x,y)=\frac{x^2y}{x^2+y^2}$ with $f_1(0,0)=0$

$f_2(x,y)=\frac{2xy^3}{(x^2+y^2)^2}$ with $f_2(0,0)=0$

In order to check for continuity at point (0,0) I know to approaches:

1) express x,y in a different coordinate system. I.e. expressing x,y in polar coordinates tells me that $f_1$ is continuous and $f_2$ is not.

2) searching for constant $C$ so that $f < xC$ or $f < yC$

However what can I try if these two criteria are inconclusive? How do you check for continuity of a multidimensional function?

Thanks in advance

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If little tricks are inconclusive, fall back on the definition (you know, $\epsilon$ and $\delta$. –  GEdgar Jun 18 '11 at 12:34
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2 Answers

up vote 2 down vote accepted

for showing continuity, you need to make some argument like

$$\Bigg|\frac{x^2y}{x^2+y^2}\Bigg|\leq\Bigg|\frac{(x^2+y^2)y}{x^2+y^2}\Bigg|=|y|$$ so as $x,y\to0$, $f_1\to0$

for showing discontinuity, just provide evidence such as $f_2(x,x)=1/2$ for $x\neq0$ so the limit cant be zero

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There is an algorithm for multiplying (decimal) numbers or polynomials, and there is an algorithm for differentiating elementary function terms; but there is no algorithm that accepts a term $f(x)$ and a limit point $\xi$ as input and produces the value of $\lim_{x\to\xi} f(x)$ as output. Depending on the circumstances, there are various stratagems to try, e.g., de l'Hôpital's rule, and so on. In the case at hand a natural thing to do is the following (you have already hinted at it): If $f(0)=0$ then introduce polar coordinates and try to get an estimate on the function $$g(r):=\sup_\phi \bigl| f(r\cos\phi, r\sin\phi)\bigr|\qquad(r>0)\ .$$ Then your function $f$ is continuous at $(0,0)$ iff $\lim_{r\to 0+} g(r)=0$.

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