How to find an uncountable set $S$, and construct an function $f : 2^S \longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in T$?

for example, let $S =\mathbb{R}$, how can I construct a funciton $f$, such that for every set $T \subseteq \mathbb{R}$, $f \left( T \right) \in T$?

we can find some wrong example:

let $T =$ {1.9, 1.99, 1.999, 1.9999, ....}

observe that if $F \left( T \right) = \max \left( T \right)$, then $\max \left( T \right) \notin T$

or $F \left( T \right) = \sup \left( T \right)$, but $\sup \left( T \right) = 2 \notin T$

Maybe the well-ordering theorem is helpful, T can be well-ordered, so we can
find an least element $t \in T$. However, this is not a constructive function,
we don't know what the element $t$ is.

So how can I **"construct"** a function satisfies the conditions stated above?