Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$f:\mathbb{R}^2\rightarrow\mathbb{R}, \ f(x,y)=x\cdot\mathbb{D}(y)$, 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..

share|cite|improve this question
Dirichlet function is bounded, now can you use the squeeze theorem for the first two limits? – Keivan May 24 '12 at 21:38
up vote 0 down vote accepted

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?

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.