$$f: \displaystyle{R\to[0,1]}$$
I can't come up with one, can anyone else ? :D
How about : $$f(x)=\begin{cases} x, x \in Q \\ 0, x\in I \end {cases}$$ ??
2026-04-06 14:47:16.1775486836
Can anyone come up with an example of a monotonously not-falling function whose breakpoints are everywhere dense set on $[0,1]$?
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What about
$$ f(x)=\sum_{y\in \Bbb{Q}, y\leq x} v_y, $$
where the $v_y$ are positive with $\sum_{y\in \Bbb{Q}} v_y<\infty$?