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Here is my question.

Suppose $f$ is a real function. Then the set $E=\{x\in \mathbb{R}:\lim\limits_{y\to x}f(y)=+\infty\}$ is countable.

I want to define an injective map from $E$ to $\mathbb{Q}$ or $\mathbb{Q}^n$, but I do not have any good ideas. I would appreciate a lot if someone can help me. Thx~

Zhang Hao
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1 Answers1

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HINT:

Show that for each $n\in \mathbb{N}$ the set $E_n\colon = E\cap f^{-1}((-\infty, n))$ is countable.

Take $x\in E_n$ and $I_x$ a symmetric open interval around $x$ so that $f>n$ on $I_x \backslash \{x\}$. Now, $I_x \not\ni y$ if $x$, $y$ in $E_n$ are distinct. Therefore, the smaller intervals $I'_x$ (half the size of $I_x$) do not intersect. It should be easy now.

orangeskid
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  • This is not rigorous enough since you have not specified the size of the intervals. $y$ could be really close to $I_x$. Moreover, you mean $E_n = E\cap f^{-1}((-\infty,n))$. – amsmath Sep 08 '19 at 02:57
  • @amsmath: $I'_x$ is open, symmetric around $x$, of half the size of $I'_x$. That guarantees that the different $I'_x$ do not intersect – orangeskid Sep 08 '19 at 03:01
  • I see now, thanks. However, you should correct your $E_n$. – amsmath Sep 08 '19 at 03:03
  • You should read my first comment. – amsmath Sep 08 '19 at 04:21