# 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) 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?

6 years, 3 months ago

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