Countably Infinite Peaks?

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

For each δ>0\delta>0, the set of all xRx\in\mathbb{R} such that f(y)<f(x)f(y)<f(x) for all yy with 0<xy<δ0<|x-y|<\delta is countable. This can be seen by noting that the set contains at most one element of the interval [kδ2,(k+1)δ2][k\frac{\delta}{2},(k+1)\frac{\delta}{2}] for each integer kk, and these intervals cover R\mathbb{R}. The set of strict local maxima is a countable union of such sets, for example taking δ=1n\delta=\frac{1}{n} as nn 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 N\mathbb{N}

Thoughts?

Note by Agnishom Chattopadhyay
4 years, 11 months ago

No vote yet
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