1
$\begingroup$

I recently had a bit of an obsession with the definition of limits, and that made think about this function: $$f: \mathbb{Q} \to \mathbb{R}; \ \ \ \ f(x) = x$$

What is $\ \lim_{x \to 0} f(x)\ $? Intuitively, I think $\ \lim_{x \to 0} f(x) = 0\ $. I then looked at the definition provided by Wikipedia: enter image description here

According to this definition, $\ \lim_{x \to 0} f(x)\ $ does not exist; $\mathbb{Q}$ does not contain within it any open intervals whatsoever, let alone ones required by the definition. However, there is also a definition for metric spaces:

enter image description here

According to this definition, the limit is indeed $\ \lim_{x \to 0} f(x) = 0\ $; setting $X, Y, N = \mathbb{R}$ and $M=\mathbb{Q}$ makes showing that result trivial. This got me wondering, why does there seem to be a requirement for an open interval (or open ball) in the first definition, and not the second?

The idea that I heard was that, the first definition has that restriction to select for only well behaving functions. The second definition captures all functions that converge to a limit, even ones that don't behave well. That seems to make sense.

I then scroll down to the topological definition, I see the following: enter image description here

This too seems to agree with the metric space definition, because with $\Omega = \mathbb{Q}\ $, $\ U \cap \Omega$ eliminates any non-rational numbers by definition. However, I also heard that (for the topological definition) there should exist such an open set $Op \subset \mathbb{R}$, so that $0 \in Op$ and $\mathbb{Q}$ contains $Op \setminus \{ 0 \}$.

I don't know if this requirement is accurate (I heard it on reddit). Is this requirement accurate?

If it is accurate, I have a follow-up question. With the standard topology on $\mathbb{R}$, the subspace $\mathbb{Q}$ itself is not an open set, and neither can it contain within it any open sets. So what gives? How is it that the requirement for $\mathbb{Q}$ containing an open set is still met, somehow?

$\endgroup$
2
  • 2
    $\begingroup$ You should not necessarily trust what you read on Wikipedia since anybody can edit no matter how little do they know about the subject, textbooks are better as a source. The general topological and metric definitions of limits are fine. The fact that $\mathbb Q$ is not open in $\mathbb R$ is irrelevant. $\endgroup$ Mar 1 at 16:02
  • 1
    $\begingroup$ In fact, if you continue reading Wikipedia's article on limits of functions (section "More general definition using limit points and subsets") you will get to the definition consistent with the topological one. $\endgroup$ Mar 2 at 1:49

0

You must log in to answer this question.

Browse other questions tagged .