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Determine all points on which the following function is continuous:

$$f(x,y) = \begin{cases} \cos\left(\frac{1}{x^2+y^2}\right) & \text{if } (x,y) \neq (0,0) \\ 1 & \text{if } (x,y) = (0,0). \end{cases}$$

By composition of functions, the function of concern is $\frac{1}{x^{2}+y^{2}}$.
In order to establish continuity we must determine the limit of $f(x,y)$ as $(x,y) \rightarrow 0$.

$$\lim_{(x,y) \to (0,0)} \frac{1}{x^{2}+y^{2}} $$

This is where I'm struggling. In most of the examples in the book, the fractions will have a variable in the numerator with which you can cancel out some element in the denominator. This will either allow you to easily find (a) some examples where you can show the limit of the function reaching different values and thus, the non-existence of the limit. Or (b) some function bigger than or equal to the function of interest that will allow you to determine the actual limits function.

I'm not sure what I can do with this limit. Ideas/suggestions/comments/critiques welcome.

Adam
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Certainly the only point where continuity may fail is $(0,0)$. To see that $f$ is discontinuous at that point, consider allowing $(x,y)$ to approach $(0,0)$ along the line $y = 0$. Then $f(x,0) = \cos(1/x^2)$, which oscillates between $\pm 1$ infinitely often as $x \rightarrow 0$.

To be precise, if $n \in \mathbb{N}$ and $$x_n = \sqrt{\frac{2}{n\pi}}$$ then $$\frac{1}{x_n^2} = \frac{n\pi}{2}$$ so $$\cos\left(\frac{1}{x_n^2}\right) = \begin{cases} 1 & \text{if }n = 1, 5, 9, \ldots \\ -1 & \text{if }n = 3, 7, 11, \ldots \\ \end{cases}$$

Therefore $\lim_{x \rightarrow 0} f(x,0)$ does not exist, so $\lim_{(x,y) \rightarrow (0,0)}f(x,y)$ does not exist. As a result, $f$ cannot be continuous at $(0,0)$.