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$f:\mathbb{R}^2\rightarrow\mathbb{R}, \ f(x,y)=x\cdot\mathbb{D}(y)$, where $\mathbb{D}$ is Dirichlet function (nowhere continuous function). Find all the limits: $\lim_{x\to 0}\lim_{y\to 0}f(x,y)$, $\lim_{y\to 0}\lim_{x\to 0}f(x,y)$ and $\lim_{(x,y)\to (0,0)}f(x,y)$.

Is there any problem with moving $x,y$ to $0$ (no matter in what order)? I think there is no problem and all limits will be equal to $0$, but then this exercise will be rather pointless, so I'm not sure..

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Dirichlet function is bounded, now can you use the squeeze theorem for the first two limits? –  Keivan May 24 '12 at 21:38

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This is a nice exercise which illustrates that the existence of a double limit does not imply the existence of iterated limits. Using the squeeze theorem (as @Keivan indicated) you should be able to prove that the 2nd and 3rd limits are zero. But the 1st one does not exist... do you see why?

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first does not exist? you mean the limit: $\lim_{x\to 0}\lim_{y\to 0}f(x,y)$? I don't know why.. –  xan May 24 '12 at 22:41
    
This iterated limit is really $\lim_{x\to 0}L(x)$ where $L(x)=\lim_{y\to 0} f(x,y)$. For the limit $\lim_{x\to 0}L(x)$ to exist, the function $L$ must be defined in some neighborhood of $0$ (except possibly $0$ itself. For which values of $x$ is $L(x)=\lim_{y\to 0} x \mathbb D(y)$ defined? –  user31373 May 24 '12 at 23:08

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