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I am curious to find if there is a characterization of the set $S$ of all functions $f: \mathbb{Q} \to \mathbb{Q}$ where for all intervals $[a,b] \subset \mathbb{Q}$, there exists $x \in [a,b]$ for every $y$ between $f(a)$ and $f(b)$ such that $f(x) = y$.

We see $f(x) = x$ has $f \in S$ since for any $y \in [f(a),f(b)] \subset \mathbb{Q}$ we can choose $x = y$ to have $f(x) = y$.

For a non example, take $f(x) = x^2$ and $[f(0),f(2)] = [0,4]$. We have $2 \in [0,4]$, but $f(x) = 2$ has no solutions. Similarly, $f(x) = x^n \not \in S$ for $n > 1$.

From the lemmas I've found so far, I conjecture $S$ is made up of piecewise defined functions of the form $a(x-b)^n+c$ for $a,b,c \in \mathbb{Q}$ and $n \in \{-1,0,1\}$ with appropreate care taken to deal with discontinuities.

Here are my questions

  • Do we miss anything restricting to piecewise rational functions $p(x)/q(x)$?
  • Supposing $f(x), g(x) \not \in S$, can we show $f(x) + g(x) \not \in S$
  • Is this well known appearing in a book somewhere?
  • Is the corresponding question for number fields well known?
Luke
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    The Conway Base 13 function (https://math.stackexchange.com/questions/1143/a-definition-of-conway-base-13-function) looks as though it takes rationals to rationals and is surjective on rationals: I can find a rational number in any interval which maps to any given rational. In both base $10$ and base $13$ rationals are characterised as having a terminating 'decimal' expansion or an expansion which ultimately repeats, and both are handled by this construction. – Mark Bennet Sep 15 '21 at 06:17
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    My suspicion is that one can recursively define an immense variety of functions with this property that have nothing to do with polynomials, rational functions, or indeed even continuous functions. See this paper of mine for some recursive constructions of order-preserving bijections of countable sets of real numbers; I believe the construction can be modified to product examples that satisfy the intermediate value property. – Greg Martin Sep 15 '21 at 06:22

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