From e1af33e8058391100082b26555639f72ba538baf Mon Sep 17 00:00:00 2001
From: Joshua Potter <jrpotter2112@gmail.com>
Date: Sat, 13 May 2023 07:14:03 -0600
Subject: [PATCH] Have definitions at top in "glossary" chapter.

---
 Bookshelf/Apostol.tex | 93 ++++++++++++++++++++++---------------------
 1 file changed, 47 insertions(+), 46 deletions(-)

diff --git a/Bookshelf/Apostol.tex b/Bookshelf/Apostol.tex
index eaf3b47..1afecaf 100644
--- a/Bookshelf/Apostol.tex
+++ b/Bookshelf/Apostol.tex
@@ -17,6 +17,53 @@
 
 \tableofcontents
 
+\chapter{Glossary}%
+\label{chap:glossary}
+
+\section{\defined{Partition}}%
+\label{sec:def-partition}
+
+Let $[a, b]$ be a closed interval decomposed into $n$ subintervals by inserting
+  $n - 1$ points of subdivision, say $x_1$, $x_2$, $\ldots$, $x_{n-1}$, subject
+  only to the restriction
+  \begin{equation}
+    \label{sec:partition-eq1}
+    a < x_1 < x_2 < \cdots < x_{n-1} < b.
+  \end{equation}
+It is convenient to denote the point $a$ itself by $x_0$ and the point $b$ by
+  $x_n$.
+A collection of points satisfying \eqref{sec:partition-eq1} is called a
+  \textbf{partition} $P$ of $[a, b]$, and we use the symbol
+  $$P = \{x_0, x_1, \ldots, x_n\}$$ to designate this partition.
+
+\begin{definition}
+
+  \lean{Common/Set/Intervals/Partition}{Set.Intervals.Partition}
+
+\end{definition}
+
+\section{\defined{Step Function}}%
+\label{sec:def-step-function}
+
+A function $s$, whose domain is a closed interval $[a, b]$, is called a step
+  function if there is a \nameref{sec:def-partition}
+  $P = \{x_0, x_1, \ldots, x_n\}$ of $[a b]$ such that $s$ is constant on each
+  open subinterval of $P$.
+That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
+  such that $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k.$$
+Step functions are sometimes called piecewise constant functions.
+
+\vspace{8pt}
+\noindent
+\textit{Note:} At each of the endpoints $x_{k-1}$ and $x_k$ the function must
+  have some well-defined value, but this need not be the same as $s_k$.
+
+\begin{definition}
+
+  \lean{Common/Set/Intervals/StepFunction}{Set.Intervals.StepFunction}
+
+\end{definition}
+
 \chapter{A Set of Axioms for the Real-Number System}%
 \label{chap:set-axioms-real-number-system}
 
@@ -1085,52 +1132,6 @@ Prove that the set function $n$ satisfies the first three axioms for area.
 
 \end{proof}
 
-\chapter{Partitions and Step Functions}%
-\label{chap:partitions-step-functions}
-
-\section{\defined{Partition}}%
-\label{sec:partition}
-
-Let $[a, b]$ be a closed interval decomposed into $n$ subintervals by inserting
-  $n - 1$ points of subdivision, say $x_1$, $x_2$, $\ldots$, $x_{n-1}$, subject
-  only to the restriction
-  \begin{equation}
-    \label{sec:partition-eq1}
-    a < x_1 < x_2 < \cdots < x_{n-1} < b.
-  \end{equation}
-It is convenient to denote the point $a$ itself by $x_0$ and the point $b$ by
-  $x_n$.
-A collection of points satisfying \eqref{sec:partition-eq1} is called a
-  \textbf{partition} $P$ of $[a, b]$, and we use the symbol
-  $$P = \{x_0, x_1, \ldots, x_n\}$$ to designate this partition.
-
-\begin{definition}
-
-  \lean{Common/Set/Intervals/Partition}{Set.Intervals.Partition}
-
-\end{definition}
-
-\section{\defined{Step Function}}%
-\label{sec:step-function}
-
-A function $s$, whose domain is a closed interval $[a, b]$, is called a step
-  function if there is a \nameref{sec:partition} $P = \{x_0, x_1, \ldots, x_n\}$
-  of $[a b]$ such that $s$ is constant on each open subinterval of $P$.
-That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
-  such that $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k.$$
-Step functions are sometimes called piecewise constant functions.
-
-\vspace{8pt}
-\noindent
-\textit{Note:} At each of the endpoints $x_{k-1}$ and $x_k$ the function must
-  have some well-defined value, but this need not be the same as $s_k$.
-
-\begin{definition}
-
-  \lean{Common/Set/Intervals/StepFunction}{Set.Intervals.StepFunction}
-
-\end{definition}
-
 \chapter{Exercises 1.11}%
 \label{chap:exercises-1-11}