Based on the following problem from http://math.uchicago.edu/~vipul/teaching-0910/151/applyingformaldefinitionoflimit.pdf:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f(x) = \begin{cases}

x^2 &, \text{ if }x\text{ is rational} \\

x &, \text{ if } x\text{ is irrational}

\end{cases}

[/tex]

is shown to have the following limit:

[tex]

\lim_{x\to 1}f(x) = 1

[/tex]

by showing ##\lim_{x\to 1}x^2 = 1## where ##x \in \mathbb{Q}## and ##\lim_{x\to 1}x = 1## where ##x \in \overline{\mathbb{Q}}## separately.

So, I suspect that limit doesnotrequire a gapless domain because the set ℚ of rational numbers as well as the set ##\overline{\mathbb{Q}}## of irrational numbers have gaps.

But, why the common ϵ-δ definition of a limit ##\lim_{x\to a}f(x)=L## (##a## is neither −∞ nor ∞)neveremphasizes the point that ##x## does not need to come from a gapless domain? For example, my https://www.amazon.com/dp/0073532320 simply says ##x## without explaining any property about ##x##, and therefore, I was wrongfully led to believe that ##x## must come from a gapless domain.

I see that Wikipedia's article on precise definition of limit explicitly states that ##x \in D \subseteq \mathbb{R}##. But, I think it may also wrongfully lead people to believe that limit ##\lim_{x\to a}f(x)=L## also works when the domain is so full of gaps like the set ##\mathbb{Z}## of integers and ##a## is neither −∞ nor ∞.

Now I am wondering: what is the precise definition of limit ##\lim_{x\to a}f(x)=L## (##a## is neither −∞ nor ∞) especially when it comes to being precise about the ##x##?

Thank you very much.

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# Limit epsilon-delta definition vs. continuity

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