I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might somehow violate continuity.
2026-03-28 08:49:28.1774687768
$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?
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Take a closed uncountable set without interior points, such as the Cantor set $C$. The distance function $$f(x) = \min\{|x-y|:y\in C\}$$ is continuous, not constant on any interval, and has a minimum at every point of $C$.