# Countably Infinite Peaks?

Some theorem states that there is no function with uncountably many strict extremal points.

For each $$\delta>0$$, the set of all $$x\in\mathbb{R}$$ such that $$f(y)<f(x)$$ for all $$y$$ with $$0<|x-y|<\delta$$ is countable. This can be seen by noting that the set contains at most one element of the interval $$[k\frac{\delta}{2},(k+1)\frac{\delta}{2}]$$ for each integer $$k$$, and these intervals cover $$\mathbb{R}$$. The set of strict local maxima is a countable union of such sets, for example taking $$\delta=\frac{1}{n}$$ as $$n$$ ranges over the positive integers.

I wonder if the peaks of the Weierstass Function could be shown to have a bijection with $$\mathbb{N}$$

Thoughts?

3 years, 7 months ago

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