| Version 8 |
Version 7 |
| \newcommand{\sR}[0]{\mathbb{R}} |
\newcommand{\sR}[0]{\mathbb{R}} |
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| \PMlinkescapeword{increasing} |
\PMlinkescapeword{increasing} |
| \PMlinkescapeword{strictly increasing} |
\PMlinkescapeword{strictly increasing} |
| \PMlinkescapeword{decreasing} |
\PMlinkescapeword{decreasing} |
| \PMlinkescapeword{strictly decreasing} |
\PMlinkescapeword{strictly decreasing} |
| \PMlinkescapeword{monotone} |
\PMlinkescapeword{monotone} |
| \PMlinkescapeword{strictly monotone} |
\PMlinkescapeword{strictly monotone} |
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| {\bf Definition} |
{\bf Definition} |
| Let $A$ be a subset of $\sR$, and let $f$ be a function from $f:A\to \sR$. |
Let $A$ be a subset of $\sR$, and let $f$ be a function from $f:A\to \sR$. |
| Then |
Then |
| \begin{enumerate} |
\begin{enumerate} |
|
\item $f$ is \emph{increasing} or \emph{weakly increasing}, if
|
\item $f$ is \emph{increasing}increasing}, if
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| $x\le y$ implies that $f(x)\le f(y)$ (for all $x$ and $y$ in $A$). |
$x\le y$ implies that $f(x)\le f(y)$ (for all $x$ and $y$ in $A$). |
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\item $f$ is \emph{strictly increasing} or \emph{strongly increasing}, if
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\item $f$ is \emph{strictly increasing}ly increasing}, if
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| $x< y$ implies that $f(x)< f(y)$. |
$x< y$ implies that $f(x)< f(y)$. |
|
\item $f$ is \emph{decreasing} or \emph{weakly decreasing}, if
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\item $f$ is \emph{decreasing}decreasing}, if
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| $x\le y$ implies that $f(x)\ge f(y)$. |
$x\le y$ implies that $f(x)\ge f(y)$. |
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\item $f$ is \emph{strictly decreasing} or \emph{strongly decreasing} if
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\item $f$ is \emph{strictly decreasing}, if
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| $x< y$ implies that $f(x)> f(y)$. |
$x< y$ implies that $f(x)> f(y)$. |
| \item $f$ is \emph{monotone}, |
\item $f$ is \emph{monotone}, |
| if $f$ is either increasing or decreasing. |
if $f$ is either increasing or decreasing. |
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\item $f$ is \emph{strictly monotone} or \emph{strongly monotone},
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\item $f$ is \emph{strictly monotone}ly monotone},
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| if $f$ is either strictly increasing or strictly decreasing. |
if $f$ is either strictly increasing or strictly decreasing. |
| \end{enumerate} |
\end{enumerate} |
|
|
| {\bf Theorem} Let $X$ be a bounded or unbounded open interval of $\sR$. |
{\bf Theorem} Let $X$ be a bounded or unbounded open interval of $\sR$. |
| In other words, let $X$ be an interval of the form $X=(a,b)$, where $a,b\in\sR\cup\{-\infty,\infty\}$. |
In other words, let $X$ be an interval of the form $X=(a,b)$, where $a,b\in\sR\cup\{-\infty,\infty\}$. |
| Futher, let $f:X\to \sR$ be a monotone function. |
Futher, let $f:X\to \sR$ be a monotone function. |
| \begin{enumerate} |
\begin{enumerate} |
| \item The set of points where $f$ is discontinuous is at most |
\item The set of points where $f$ is discontinuous is at most |
| countable \cite{aliprantis, rudin}. |
countable \cite{aliprantis, rudin}. |
| \item [Lebesgue] $f$ is differentiable almost |
\item [Lebesgue] $f$ is differentiable almost |
| everywhere (\cite{jones}, pp. 514). |
everywhere (\cite{jones}, pp. 514). |
| \end{enumerate} |
\end{enumerate} |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{aliprantis} |
\bibitem{aliprantis} |
| C.D. Aliprantis, O. Burkinshaw, \emph{Principles of Real Analysis}, |
C.D. Aliprantis, O. Burkinshaw, \emph{Principles of Real Analysis}, |
| 2nd ed., Academic Press, 1990. |
2nd ed., Academic Press, 1990. |
| \bibitem{rudin} |
\bibitem{rudin} |
| W. Rudin, \emph{Principles of Mathematical Analysis}, McGraw-Hill Inc., 1976. |
W. Rudin, \emph{Principles of Mathematical Analysis}, McGraw-Hill Inc., 1976. |
| \bibitem{jones} |
\bibitem{jones} |
| F. Jones, \emph{Lebesgue Integration on Euclidean Spaces}, |
F. Jones, \emph{Lebesgue Integration on Euclidean Spaces}, |
| Jones and Barlett Publishers, 1993. |
Jones and Barlett Publishers, 1993. |
| \end{thebibliography} |
\end{thebibliography} |
