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If $f$, for $x>0$, is a continuous function such that for all rationals $m=p/q$ with $\gcd(p,q)=1$, $f(m)$ is equal to the number of digits of $q$ (either base-2 or base-10), what is $f(\pi)$?

And 2) If $f(m)=q$, then what is $f(\pi)$?

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    I dont know. It says "And 2) I$f$ f(m)=q"2012-02-03

1 Answers 1

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Such a function cannot possibly exist. To see this, first observe

$f\left(\frac{n+1}{n}\right)=\left\lfloor \log_b \, n \right\rfloor \xrightarrow{n\to\infty}\infty.$

All the while, $(n+1)/n\xrightarrow{n\to\infty}1$ but $f(1)=1\ne\infty$, which contradicts continuity.

This same example applies to case (2) as well.