r
/
bookshelf
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Apostol. Update coloring of definitions in the reference section.

finite-set-exercises
Joshua Potter 2023-08-08 17:32:32 -06:00
parent ec53cdd710
commit 51e415a87f
1 changed files with 7 additions and 7 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

14
Bookshelf/Apostol.tex
View File

@ -65,7 +65,7 @@ Such a number $B$ is also known as the \textbf{greatest lower bound}.
\end{definition} \end{definition}
\section{\pending{Integrable}}% \section{\defined{Integrable}}%
\hyperlabel{ref:integrable} \hyperlabel{ref:integrable}
Let $f$ be a function defined and bounded on $[a, b]$. Let $f$ be a function defined and bounded on $[a, b]$.
@ -74,7 +74,7 @@ $f$ is said to be \textbf{integrable} if there exists one and only one number
If $f$ is integrable on $[a, b]$, we say that the integral If $f$ is integrable on $[a, b]$, we say that the integral
$\int_a^b f(x) \mathop{dx}$ \textbf{exists}. $\int_a^b f(x) \mathop{dx}$ \textbf{exists}.
\section{\pending{Integral of a Bounded Function}}% \section{\defined{Integral of a Bounded Function}}%
\hyperlabel{ref:integral-bounded-function} \hyperlabel{ref:integral-bounded-function}
Let $f$ be a function defined and bounded on $[a, b]$. Let $f$ be a function defined and bounded on $[a, b]$.
@ -102,7 +102,7 @@ The function $f$ is called the \textbf{integrand}, the numbers $a$ and $b$ are
called the \textbf{limits of integration}, and the interval $[a, b]$ the called the \textbf{limits of integration}, and the interval $[a, b]$ the
\textbf{interval of integration}. \textbf{interval of integration}.
\section{\pending{Integral of a Step Function}}% \section{\defined{Integral of a Step Function}}%
\hyperlabel{ref:integral-step-function} \hyperlabel{ref:integral-step-function}
Let $s$ be a \nameref{ref:step-function} defined on $[a, b]$, and let Let $s$ be a \nameref{ref:step-function} defined on $[a, b]$, and let
@ -118,7 +118,7 @@ The \textbf{integral of $s$ from $a$ to $b$}, denoted by the symbol
If $a < b$, we define $\int_b^a s(x) \mathop{dx} = -\int_a^b s(x) \mathop{dx}$. If $a < b$, we define $\int_b^a s(x) \mathop{dx} = -\int_a^b s(x) \mathop{dx}$.
We also define $\int_a^a s(x) \mathop{dx} = 0$. We also define $\int_a^a s(x) \mathop{dx} = 0$.
\section{\pending{Lower Integral}}% \section{\defined{Lower Integral}}%
\hyperlabel{ref:lower-integral} \hyperlabel{ref:lower-integral}
Let $f$ be a function bounded on $[a, b]$ and $S$ denote the set of numbers Let $f$ be a function bounded on $[a, b]$ and $S$ denote the set of numbers
@ -128,7 +128,7 @@ That is, let $$S = \left\{ \int_a^b s(x) \mathop{dx} : s \leq f \right\}.$$
The number $\sup{S}$ is called the \textbf{lower integral of $f$}. The number $\sup{S}$ is called the \textbf{lower integral of $f$}.
It is denoted as $\ubar{I}(f)$. It is denoted as $\ubar{I}(f)$.
\section{\pending{Monotonic}}% \section{\defined{Monotonic}}%
\hyperlabel{ref:monotonic} \hyperlabel{ref:monotonic}
A function $f$ is called \textbf{monotonic} on set $S$ if it is increasing on A function $f$ is called \textbf{monotonic} on set $S$ if it is increasing on
@ -164,7 +164,7 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a
\end{definition} \end{definition}
\section{\pending{Refinement}}% \section{\defined{Refinement}}%
\hyperlabel{ref:refinement} \hyperlabel{ref:refinement}
Let $P$ be a \nameref{ref:partition} of closed interval $[a, b]$. Let $P$ be a \nameref{ref:partition} of closed interval $[a, b]$.
@ -213,7 +213,7 @@ Such a number $B$ is also known as the \textbf{least upper bound}.
\end{definition} \end{definition}
\section{\pending{Upper Integral}}% \section{\defined{Upper Integral}}%
\hyperlabel{ref:upper-integral} \hyperlabel{ref:upper-integral}
Let $f$ be a function bounded on $[a, b]$ and $T$ denote the set of numbers Let $f$ be a function bounded on $[a, b]$ and $T$ denote the set of numbers