×

# 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, 1 month ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

There are no comments in this discussion.

×