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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.

http://upload.wikimedia.org/wikipedia/commons/6/60/WeierstrassFunction.svg

http://upload.wikimedia.org/wikipedia/commons/6/60/WeierstrassFunction.svg

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

Thoughts?

Note by Agnishom Chattopadhyay
3 years, 1 month ago

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