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I'm studying for my exam and I have a bit of trouble with these kind of exercices, since I have no theory and im a bit lost:

Study the continuity in (0,0) (g(x,y) is a function depending of the exercise): $$f(x,y) = \begin{cases} g(x,y)&\text{for } (x,y)\not=(0,0)\\ 0 &\text{for }(x,y) =(0,0). \end{cases}$$ I have 2 questions:

1.What steps, or methods should I try in order to find the continuity on a function with 2 variables? (Iterated limits, etc)

2.Is there any method that can make you totally sure that a function is continous, or you can only say if its discontinous?

I really appreciate any help

Edit: This is not a real exercise its a kind of exercise.

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A function $f:\mathbb{R}^2\to\mathbb{R}$ is continuous at a point $(x_0,y_0)$ if and only if $$ \lim_{(x,y)\to(x_0,y_0)}f(x,y)=f(x_0,y_0). $$ This has to be true irrespective of "how" the point $(x,y)$ approaches the point $(x_0,y_0)$. In your case, the function $f$ coincides with a function $g$ on $\mathbb{R}^2\backslash\{(0,0)\}$ so is continuous at $(x_0,y_0)\neq(0,0)$ if and only if $g$ is continuous there.

For the point $(0,0)$ you will have to show that $$ \lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}g(x,y)=f(0,0)=0. $$

Since the function $g$ is not specified, not much else can be said.

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Thanks for the reply, what methods can I use in order solve that limit with 2 variables? I read something about iterated limits and that there are more ways to solve it. – Alejandro Jan 12 '13 at 20:44
@Alejandro: The method you would use would depend on the function $g$. If by iterated limits you mean performing the limits $x\to0$ and $y\to 0$ sequentially then this is not a good idea and may give a wrong result. In general, you can always resort to the $\epsilon-\delta$ definition of limits. – Eckhard Jan 12 '13 at 20:49

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