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An rr function (i.e. rational rational function) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

QUESTION Do there exist rr functions $\ \phi\ \psi\ $ such that set

$$ \{(\phi(x),\ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of $\ \Bbb Q^2\,?$

I'd guess -- NO.

An rr function (i.e. rational rational function) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

QUESTION Do there exist rr functions $\ \phi\ \psi\ $ such that set

$$ \{(\phi(x)\ \ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of $\ \Bbb Q^2\,?$

I'd guess -- NO.

An rr function (i.e. rational rational function) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f\ g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

QUESTION Do there exist rr functions $\ \phi\ \psi\ $ such that set

$$ \{(\phi(x)\,\ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of $\ \Bbb Q^2\,?$

I'd guess -- NO.

An rr function (i.e. rational rational function) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

QUESTION Do there exist rr functions $\ \phi\ \psi\ $ such that set

$$ \{(\phi(x),\ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of $\ \Bbb Q^2\,?$

I'd guess -- NO.

An rr function (i.e. rational rational function) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f\ g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

QUESTION Do there exist rr functions $\ \phi\ \psi\ $ such that set

$$ \{(\phi(x)\ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of $\ \Bbb Q^2\,?$

I'd guess -- NO.

An rr function (i.e. rational rational function) is a quotient

$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$

such that $\ f\ g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$

QUESTION Do there exist rr functions $\ \phi\ \psi\ $ such that set

$$ \{(\phi(x)\,\ \psi(x))\,:\, x\in\Bbb Q\}\ \subseteq \ (\Bbb Q\cup\{\infty\})^2 $$

is dense in a non-empty open subset of $\ \Bbb Q^2\,?$

I'd guess -- NO.