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

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

Thoughts?

#Calculus #Infinity #Analysis #Uncountable #Maxima

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$