Is there an example of a Darboux function which is bounded on the closed interval $[0,1]$ but achieves no absolute maximum on $[0,1]$? While I am trying to find any such function, I would prefer one that is as simple as possible.
2026-04-06 02:35:10.1775442910
Bounded Darboux Function without Absolute Maximum on Closed Interval
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$f(0)=0$, $f(x)=e^{-x}\cos\frac1x$ for $0\lt x\le1$.