Is there a function $f\colon\mathbb R \to \mathbb R$, such that its limits at rational points approach infinity?
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The answer I posted, which you accepted, was wrong. I assumed that $f(x)\gt n$ in a neighborhood $U_{n,q}$ of $q,$ which is nonsense; that inequality only has to hold in a deleted neighborhood $U_{n,q}\setminus{q}.$ I have edited my answer; hope it's right now. – bof Mar 17 '17 at 12:03
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Suppose $\lim_{x\to q}f(x)=\infty$ for each $q\in\mathbb Q.$ Given $q\in\mathbb Q$ and $n\in\mathbb N,$ there is a neighborhood $U_{n,q}$ of $q$ such that $f(x)\gt n$ for all $x\in U_{n,q}\setminus\{q\}.$ For each $n\in\mathbb N,$ the set $U_n=\bigcup_{q\in\mathbb Q}U_{n,q}$ is a dense open set, and $f(x)\gt n$ for every irrational $x\in U_n.$ By the Baire category theorem, the set $$\left(\bigcap_{n\in\mathbb N}U_n\right)\setminus\mathbb Q=\left(\bigcap_{n\in\mathbb N}U_n\right)\cap\left(\bigcap_{q\in\mathbb Q}(\mathbb R\setminus\{q\})\right)$$ is nonempty, i.e., there is an irrational number $x\in\bigcap_{n\in\mathbb N}U_n.$ Then $f(x)\gt n$ for all $n\in\mathbb N,$ which is absurd.
bof
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More generally, the set of points at which a function has an infinite limit ($+\infty$ for all such points, or $-\infty$ for all such points, or an infinite limit where the sign can vary with the point) is characterized by being a scattered set (a countable $G_{\delta}$ set). A scattered set a super-strong nowhere dense set --- a nowhere dense set has the property that it's closure is not dense in any interval, whereas a scattered set has the property that it's closure is not dense in any interval or in any Cantor set. – Dave L. Renfro Mar 17 '17 at 14:16
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See Variations on continuity: sets of infinite limit by Bumcrot and Sheingorn in Mathematics Magazine 47 # 1 (January 1974), pp. 41-43. – Dave L. Renfro Mar 24 '17 at 14:02
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The Math Forum link to the sci.math post I cited in my 17 March 2017 comment is no longer valid, but the google sci.math archive is still available -- see this 26 May 2006 sci.math post. I'm not sure if this is the actual post my previous comment cited, but this post along with other posts in the same sci.math thread certainly provide some pointers to the literature. See also this 4 May 2010 sci.math post. – Dave L. Renfro Aug 01 '23 at 20:03