Does this set have a maximum?

Question

Consider the function $f:[0,1]\longrightarrow\mathbb{R}$ where \begin{equation} f(x) = \left\{ \begin{array}{ll} 0 & \mathrm{if\ } x \in \mathbb{R-Q}\\ 1/q & \mathrm{if\ } x \in \mathbb{Q}\ \text{ and }\ x=p/q\ \text{ in lowest terms} \end{array} \right. \end{equation} Now let $P=\{t_{0},\ldots,t_{n}\}$ be a partition of $[0,1]$. I want to know if the set $A_{i}=\{f(x):t_{i-1}\leq x\leq t_{i}\}$ has a maximum for each $i=1,\ldots,n$.

Answer 1

There is a (rather trivial) order-isomorphism between $(\omega+1, <)$ and $(\{\frac1n \mid n \in \Bbb N_{>0}\}, >)$, i.e. a bijection $\varphi$ from $\omega+1$ to the image of $f$ such that $\varphi(x) > \varphi(y)$ whenever $x < y$.

Since the former is a well-ordering, i.e. every non-empty set has a minimum, so in the latter set, every non-empty set has a maximum.

Since every $A_i$ is non-empty, every $A_i$ has a maximum.

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In simpler terms, it means that check whether $1 \in A_i$, and then whether $\frac12 \in A_i$, etc. and if everything is not in $A_i$, then $0$ is the maximum.

(The latter case cannot occur for this function since every interval contains a rational, but it does not matter).