52 lines
1.6 KiB
TeX
52 lines
1.6 KiB
TeX
\documentclass[../script.tex]{subfiles}
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%! TEX root = ../../script.tex
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\begin{document}
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\section{Limits and Functions}
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In this chapter we will introduce the notation
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\[
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\oball(x) = (x - \epsilon, x + \epsilon)
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\]
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\begin{defi}
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Let $D \subset \realn$ and $x \in \realn$. $x$ is called a boundary point of $D$ if
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\[
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\forall \epsilon > 0: ~~D \cap \oball(x) \ne 0
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\]
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The set of all boundary points of $D$ is called closure and is denoted as $\closure{D}$.
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\end{defi}
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\begin{eg}
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\begin{enumerate}[(i)]
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\item $x \in D$ is always a boundary point of $D$, because
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\[
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x \in D \cap \oball(x)
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\]
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\item Boundary points don't have to be elements of $D$. If $D = (0, 1)$, then $0$ and $1$ are boundary points, because
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\[
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\frac{\epsilon}{2} \in (0, 1) \cap \oball(0) = (-\epsilon, \epsilon) ~~\forall \epsilon > 0
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\]
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\item Let $D = \ratn$. Every $x \in \realn$ is a boundary point, because $\forall \epsilon > 0$, $\oball(x)$ contains at least one rational number. I.e. $\closure{\ratn} = \realn$.
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\end{enumerate}
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\end{eg}
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\begin{rem}
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If $x$ is a boundary point, then
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\[
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\forall \epsilon > 0 ~\exists y \in D: ~~|x - y| < \epsilon
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\]
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If $x$ is not a boundary point, then
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\[
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\exists \epsilon > 0 ~\forall y \in D: ~~|x - y| \ge \epsilon
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\]
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\end{rem}
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\begin{thm}
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\[
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x \in \realn \text{ is a boundary point of } D \subset \realn \iff \exists \anyseqdef{D} \text{ such that } x_n \rightarrow x
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\]
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\end{thm}
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\end{document} |