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-<?xml version="1.0" encoding="UTF-8"?>
-<?latexml searchpaths=".,../../sty,../../rnc,../../rnc/omdoc,../../sty/modules,../../sty/statements,../../sty/sproof,../../sty/omtext,../../sty/omdoc,../../sty/sref,../../sty/presentation,../../sty/dcm,../../sty/reqdoc,../../sty/omd,../../sty/mikoslides,../../sty/problem,../../sty/assignment,../../sty/etc,/Users/kohlhase/stc/stex/example/background,../../sty,../../rnc,../../rnc/omdoc,../../sty/modules,../../sty/statements,../../sty/sproof,../../sty/omtext,../../sty/omdoc,../../sty/sref,../../sty/presentation,../../sty/dcm,../../sty/reqdoc,../../sty/omd,../../sty/mikoslides,../../sty/problem,../../sty/assignment,../../sty/etc"?>
-<?latexml class="omdoc"?>
-<?latexml package="stex,amssymb"?>
-<?latexml package="alltt"?>
-<?latexml package="hyperref"?>
-<?latexml package="listings"?>
-<!-- %% defining the author metadata -->
-<?latexml RelaxNGSchema="omdoc+ltxml"?>
-<omdoc xmlns="http://omdoc.org/ns" xmlns:stex="http://kwarc.info/ns/sTeX" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:om="http://www.openmath.org/OpenMath" xmlns:omdoc="http://omdoc.org/ns" xmlns:m="http://www.w3.org/1998/Math/MathML" stex:srcref="paper.tex#textrange(from=15;1,to=15;17)" xml:id="paper.omdoc">
- <!-- %metadata and title page -->
- <metadata xml:id="paper.omdoc.p1" about="#paper.omdoc.p1" stex:srcref="paper.tex#textrange(from=18;0,to=25;16)">
- <dc:creator xml:id="paper.omdoc.p1.p1" about="#paper.omdoc.p1.p1" stex:srcref="paper.tex#textrange(from=19;0,to=19;20)">
- <p xml:id="paper.omdoc.p1.p1.p1" about="#paper.omdoc.p1.p1.p1" stex:srcref="paper.tex#textrange(from=19;0,to=19;20)">Michael Kohlhase</p>
- </dc:creator>
- <dc:rights xml:id="paper.omdoc.p1.p2" about="#paper.omdoc.p1.p2" stex:srcref="paper.tex#textrange(from=20;0,to=20;49)">
- <p xml:id="paper.omdoc.p1.p2.p1" about="#paper.omdoc.p1.p2.p1" stex:srcref="paper.tex#textrange(from=20;0,to=20;49)">Copyright (c) 2009 Michael Kohlhase</p>
- </dc:rights>
- <dc:title xml:id="paper.omdoc.p1.p3" about="#paper.omdoc.p1.p3" stex:srcref="paper.tex#textrange(from=21;0,to=21;53)">
- <p xml:id="paper.omdoc.p1.p3.p1" about="#paper.omdoc.p1.p3.p1" stex:srcref="paper.tex#textrange(from=21;0,to=21;53)">An example of semantic Markup in <phrase xml:id="paper.omdoc.p1.p3.p1.p1" about="#paper.omdoc.p1.p3.p1.p1" stex:srcref="paper.tex#textrange(from=21;0,to=21;53)">sTeX</phrase></p>
- </dc:title>
- <dc:description xml:id="paper.omdoc.p1.p4" about="#paper.omdoc.p1.p4" stex:srcref="paper.tex#textrange(from=24;0,to=24;40)">
- <p xml:id="paper.omdoc.p1.p4.p1" about="#paper.omdoc.p1.p4.p1" stex:srcref="paper.tex#textrange(from=24;0,to=24;40)">In this note we give an example of semantic markup in <phrase xml:id="paper.omdoc.p1.p4.p1.p1" about="#paper.omdoc.p1.p4.p1.p1" stex:srcref="paper.tex#textrange(from=24;0,to=24;40)">sTeX</phrase>:
-Continuous and differentiable functions are introduced using real numbers, sets and
-functions as an assumed background.</p>
- </dc:description>
- <!-- %**** paper.tex Line 25 **** -->
- </metadata>
- <omgroup layout="sectioning" xml:id="intro" about="#intro" stex:srcref="paper.tex#textrange(from=27;0,to=110;13)">
- <metadata xml:id="intro.p1" about="#intro.p1" stex:srcref="paper.tex#textrange(from=27;0,to=27;39)">
- <dc:title stex:srcref="paper.tex#textrange(from=27;0,to=110;13)" xml:id="intro.p1.p1" about="#intro.p1.p1">
- <p xml:id="intro.p1.p1.p1" about="#intro.p1.p1.p1" stex:srcref="paper.tex#textrange(from=27;0,to=110;13)">Introduction</p>
- </dc:title>
- </metadata>
- <omtext xml:id="intro.p2" about="#intro.p2" stex:srcref="paper.tex#textrange(from=27;0,to=63;13)">
- <CMP xml:id="intro.p2.p1" about="#intro.p2.p1" stex:srcref="paper.tex#textrange(from=27;0,to=63;13)">
- <p xml:id="intro.p2.p1.p1" about="#intro.p2.p1.p1" stex:srcref="paper.tex#textrange(from=27;0,to=27;39)">We will show how to semantically mark up mathematics in the <phrase xml:id="intro.p2.p1.p1.p1" about="#intro.p2.p1.p1.p1" stex:srcref="paper.tex#textrange(from=27;1,to=28;68)">sTeX</phrase>
-format <cite xmlns="http://dlmf.nist.gov/LaTeXML">[<bibref bibrefs="kohlhase:ulsmf08" separator="," show="Number" yyseparator=","/>]</cite> and how to convert it into
-OMDoc <cite xmlns="http://dlmf.nist.gov/LaTeXML">[<bibref bibrefs="kohlhase:omdoc1.3" separator="," show="Number" yyseparator=","/>]</cite>.
-</p>
- <p xml:id="intro.p2.p1.p2" about="#intro.p2.p1.p2" stex:srcref="paper.tex#textrange(from=27;0,to=63;13)">We have some mathematical material in Section <ref type="cite" xref="sec.math" xml:id="intro.p2.p1.p2.p1" about="#intro.p2.p1.p2.p1" stex:srcref="paper.tex#textrange(from=31;29,to=32;56)"/> which will serve as example
-content. In the rest of this section we will explain the setup of the example and
-present an approach to automation of the OMDoc conversion via Unix Makefiles.
-</p>
- </CMP>
- </omtext>
- <omgroup layout="sectioning" xml:id="intro.p3" about="#intro.p3" stex:srcref="paper.tex#textrange(from=35;0,to=63;13)">
- <metadata xml:id="intro.p3.p1" about="#intro.p3.p1" stex:srcref="paper.tex#textrange(from=35;0,to=62;13)">
- <dc:title stex:srcref="paper.tex#textrange(from=35;0,to=63;13)" xml:id="intro.p3.p1.p1" about="#intro.p3.p1.p1">
- <p xml:id="intro.p3.p1.p1.p1" about="#intro.p3.p1.p1.p1" stex:srcref="paper.tex#textrange(from=35;0,to=63;13)">The Setup</p>
- </dc:title>
- </metadata>
- <theory xml:id="setup" about="#setup" stex:srcref="paper.tex#textrange(from=36;0,to=62;13)">
- <symbol name="target" xml:id="target.def.sym" about="#target.def.sym" stex:srcref="paper.tex#textrange(from=37;0,to=44;20)"/>
- <symbol name="module" xml:id="module.def.sym" about="#module.def.sym" stex:srcref="paper.tex#textrange(from=37;0,to=44;20)"/>
- <definition for="target module" xml:id="setup.p3" about="#setup.p3" stex:srcref="paper.tex#textrange(from=37;0,to=44;20)">
- <CMP xml:id="setup.p3.p1" about="#setup.p3.p1" stex:srcref="paper.tex#textrange(from=37;0,to=44;20)">
- <p xml:id="setup.p3.p1.p1" about="#setup.p3.p1.p1" stex:srcref="paper.tex#textrange(from=37;0,to=44;20)">
-The source of this note is contained in the file <phrase class="code" xml:id="setup.p3.p1.p1.p1" about="#setup.p3.p1.p1.p1" stex:srcref="paper.tex#textrange(from=37;13,to=38;74)">paper.tex</phrase>. We call it
-the <idx xml:id="setup.p3.p1.p1.p2" about="#setup.p3.p1.p1.p2" stex:srcref="paper.tex#textrange(from=38;59,to=39;25)"><idt><term cd="setup" name="target" role="definiendum" xml:id="setup.p3.p1.p1.p2.p1" about="#setup.p3.p1.p1.p2.p1" stex:srcref="paper.tex#textrange(from=38;59,to=39;25)">target</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">target</text></idp></ide></idx>, since formatting it with LaTeX will generate the main
-document. The content in Section <ref type="cite" xref="sec.contfuncs" xml:id="setup.p3.p1.p1.p3" about="#setup.p3.p1.p1.p3" stex:srcref="paper.tex#textrange(from=39;33,to=40;52)"/> and Section <ref type="cite" xref="sec.differentiable" xml:id="setup.p3.p1.p1.p4" about="#setup.p3.p1.p1.p4" stex:srcref="paper.tex#textrange(from=39;1,to=40;84)"/>
-comes from included files <phrase class="code" xml:id="setup.p3.p1.p1.p5" about="#setup.p3.p1.p1.p5" stex:srcref="paper.tex#textrange(from=40;5,to=41;56)">continuous.tex</phrase> and
-<phrase class="code" xml:id="setup.p3.p1.p1.p6" about="#setup.p3.p1.p1.p6" stex:srcref="paper.tex#textrange(from=41;54,to=42;34)">differentiable.tex</phrase>, we will call them <idx xml:id="setup.p3.p1.p1.p7" about="#setup.p3.p1.p1.p7" stex:srcref="paper.tex#textrange(from=41;18,to=42;70)"><idt><term cd="setup" name="module" role="definiendum" xml:id="setup.p3.p1.p1.p7.p1" about="#setup.p3.p1.p1.p7.p1" stex:srcref="paper.tex#textrange(from=41;18,to=42;70)">module</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">module</text></idp></ide></idx>s, since they may
-be used (i.e. included) by other target documents as well.
-</p>
- </CMP>
- </definition>
- <omtext xml:id="setup.p4" about="#setup.p4" stex:srcref="paper.tex#textrange(from=36;0,to=54;19)">
- <CMP xml:id="setup.p4.p1" about="#setup.p4.p1" stex:srcref="paper.tex#textrange(from=36;0,to=54;19)">
- <p xml:id="setup.p4.p1.p1" about="#setup.p4.p1.p1" stex:srcref="paper.tex#textrange(from=36;0,to=54;19)">As the modules are built for inclusion into other documents, they are not
-self-contained:
-</p>
- </CMP>
- </omtext>
- <omgroup layout="enumerate" xml:id="I1" about="#I1" stex:srcref="paper.tex#textrange(from=47;0,to=54;19)">
- <omtext xml:id="I1.p1" about="#I1.p1" stex:srcref="paper.tex#textrange(from=47;42,to=48;11)">
- <CMP xml:id="I1.p1.p1" about="#I1.p1.p1" stex:srcref="paper.tex#textrange(from=47;42,to=48;11)">
- <p xml:id="I1.p1.p1.p1" about="#I1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=47;42,to=48;11)">they do not contain a LaTeX preamble and
-<verbatim xmlns="http://dlmf.nist.gov/LaTeXML" font="typewriter">\begin</verbatim>/<verbatim xmlns="http://dlmf.nist.gov/LaTeXML" font="typewriter">\end{document}</verbatim>, and
-<!-- %**** paper.tex Line 50 **** --></p>
- </CMP>
- </omtext>
- <omtext xml:id="I1.p2" about="#I1.p2" stex:srcref="paper.tex#textrange(from=49;76,to=50;11)">
- <CMP xml:id="I1.p2.p1" about="#I1.p2.p1" stex:srcref="paper.tex#textrange(from=49;76,to=50;11)">
- <p xml:id="I1.p2.p1.p1" about="#I1.p2.p1.p1" stex:srcref="paper.tex#textrange(from=49;76,to=50;11)">they may depend on other modules, whose semantic macros they need to include,
-</p>
- </CMP>
- </omtext>
- <omtext xml:id="I1.p3" about="#I1.p3" stex:srcref="paper.tex#textrange(from=50;75,to=51;11)">
- <CMP xml:id="I1.p3.p1" about="#I1.p3.p1" stex:srcref="paper.tex#textrange(from=50;75,to=51;11)">
- <p xml:id="I1.p3.p1.p1" about="#I1.p3.p1.p1" stex:srcref="paper.tex#textrange(from=50;75,to=51;11)">to facilitate this a module file <phrase class="code" xml:id="I1.p3.p1.p1.p1" about="#I1.p3.p1.p1.p1" stex:srcref="paper.tex#textrange(from=50;25,to=51;61)">modf.tex</phrase> comes with a “semantic
-macro short form” <phrase class="code" xml:id="I1.p3.p1.p1.p2" about="#I1.p3.p1.p1.p2" stex:srcref="paper.tex#textrange(from=51;41,to=52;43)">modf.sms</phrase> that can be included without generating
-output in the PDF.
-</p>
- </CMP>
- </omtext>
- </omgroup>
- <symbol name="background-module" xml:id="background-module.def.sym"/>
- <omtext xml:id="setup.p6" about="#setup.p6" stex:srcref="paper.tex#textrange(from=36;0,to=62;13)">
- <CMP xml:id="setup.p6.p1" about="#setup.p6.p1" stex:srcref="paper.tex#textrange(from=36;0,to=62;13)">
- <p xml:id="setup.p6.p1.p1" about="#setup.p6.p1.p1" stex:srcref="paper.tex#textrange(from=36;0,to=62;13)">this will have consequences for the automation. Concretely, the module on
-differentiable functions in Section <ref type="cite" xref="sec.differentiable" xml:id="setup.p6.p1.p1.p1" about="#setup.p6.p1.p1.p1" stex:srcref="paper.tex#textrange(from=55;32,to=56;58)"/> depends on that for continuous
-functions in Section <ref type="cite" xref="sec.contfuncs" xml:id="setup.p6.p1.p1.p2" about="#setup.p6.p1.p1.p2" stex:srcref="paper.tex#textrange(from=56;51,to=57;38)"/>. Both of them depend on modules for real numbers,
-sets and functions that we do not want to cover in this note. <phrase for="background-module" type="inlinedef" xml:id="setup.p6.p1.p1.p3" about="#setup.p6.p1.p1.p3" stex:srcref="paper.tex#textrange(from=59;17,to=60;69)">We assume that
-they have already been marked up with the same methods as we describe here and are
-accessible to us and call them <idx xml:id="setup.p6.p1.p1.p3.p1" about="#setup.p6.p1.p1.p3.p1" stex:srcref="paper.tex#textrange(from=59;17,to=60;69)"><idt><term cd="setup" name="background-module" role="definiendum" xml:id="setup.p6.p1.p1.p3.p1.p1" about="#setup.p6.p1.p1.p3.p1.p1" stex:srcref="paper.tex#textrange(from=59;17,to=60;69)">background module</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">background</text></idp><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">module</text></idp></ide></idx>s</phrase>. In our setup we
-keep them in the subdirectory <phrase class="code" xml:id="setup.p6.p1.p1.p4" about="#setup.p6.p1.p1.p4" stex:srcref="paper.tex#textrange(from=60;2,to=61;54)">background</phrase>.
-</p>
- </CMP>
- </omtext>
- </theory>
- </omgroup>
- <omgroup layout="sectioning" xml:id="intro.p4" about="#intro.p4" stex:srcref="paper.tex#textrange(from=65;0,to=73;13)">
- <metadata xml:id="intro.p4.p1" about="#intro.p4.p1" stex:srcref="paper.tex#textrange(from=65;0,to=65;51)">
- <dc:title stex:srcref="paper.tex#textrange(from=65;0,to=73;13)" xml:id="intro.p4.p1.p1" about="#intro.p4.p1.p1">
- <p xml:id="intro.p4.p1.p1.p1" about="#intro.p4.p1.p1.p1" stex:srcref="paper.tex#textrange(from=65;0,to=73;13)">Formatting and OMDoc conversion</p>
- </dc:title>
- </metadata>
- <omtext xml:id="intro.p4.p2" about="#intro.p4.p2" stex:srcref="paper.tex#textrange(from=65;0,to=73;13)">
- <CMP xml:id="intro.p4.p2.p1" about="#intro.p4.p2.p1" stex:srcref="paper.tex#textrange(from=65;0,to=73;13)">
- <p xml:id="intro.p4.p2.p1.p1" about="#intro.p4.p2.p1.p1" stex:srcref="paper.tex#textrange(from=65;0,to=65;51)">To format an <phrase xml:id="intro.p4.p2.p1.p1.p1" about="#intro.p4.p2.p1.p1.p1" stex:srcref="paper.tex#textrange(from=65;68,to=66;21)">sTeX</phrase> document — i.e. to produce a PDF file from the <phrase xml:id="intro.p4.p2.p1.p1.p2" about="#intro.p4.p2.p1.p1.p2" stex:srcref="paper.tex#textrange(from=65;11,to=66;78)">sTeX</phrase> marked-up
-sources — we only need to run the <phrase class="code" xml:id="intro.p4.p2.p1.p1.p3" about="#intro.p4.p2.p1.p1.p3" stex:srcref="paper.tex#textrange(from=66;34,to=67;56)">pdflatex</phrase> program over the target document
-— assuming that all modules (regular or background) have semantic macro short forms.
-</p>
- <p xml:id="intro.p4.p2.p1.p2" about="#intro.p4.p2.p1.p2" stex:srcref="paper.tex#textrange(from=65;0,to=73;13)">To convert an <phrase xml:id="intro.p4.p2.p1.p2.p1" about="#intro.p4.p2.p1.p2.p1" stex:srcref="paper.tex#textrange(from=69;66,to=70;22)">sTeX</phrase> document to OMDoc, we need to run <phrase class="code" xml:id="intro.p4.p2.p1.p2.p2" about="#intro.p4.p2.p1.p2.p2" stex:srcref="paper.tex#textrange(from=69;10,to=70;78)">latexml</phrase> over it,
-post-process the result by <phrase class="code" xml:id="intro.p4.p2.p1.p2.p3" about="#intro.p4.p2.p1.p2.p3" stex:srcref="paper.tex#textrange(from=70;41,to=71;50)">latexmlpost</phrase>, and finally massage away all remaining
-LaTeXML islands with a stylesheet, see  <cite xmlns="http://dlmf.nist.gov/LaTeXML">[<bibref bibrefs="kohlhase:ssmtl*" separator="," show="Number" yyseparator=","/>]</cite> for details.
-</p>
- </CMP>
- </omtext>
- </omgroup>
- <!-- %**** paper.tex Line 75 **** -->
- <omgroup layout="sectioning" xml:id="intro.p5" about="#intro.p5" stex:srcref="paper.tex#textrange(from=75;0,to=108;13)">
- <metadata xml:id="intro.p5.p1" about="#intro.p5.p1" stex:srcref="paper.tex#textrange(from=75;0,to=75;42)">
- <dc:title stex:srcref="paper.tex#textrange(from=75;0,to=108;13)" xml:id="intro.p5.p1.p1" about="#intro.p5.p1.p1">
- <p xml:id="intro.p5.p1.p1.p1" about="#intro.p5.p1.p1.p1" stex:srcref="paper.tex#textrange(from=75;0,to=108;13)">Makefile-based Automation</p>
- </dc:title>
- </metadata>
- <omtext xml:id="intro.p5.p2" about="#intro.p5.p2" stex:srcref="paper.tex#textrange(from=75;0,to=87;19)">
- <CMP xml:id="intro.p5.p2.p1" about="#intro.p5.p2.p1" stex:srcref="paper.tex#textrange(from=75;0,to=87;19)">
- <p xml:id="intro.p5.p2.p1.p1" about="#intro.p5.p2.p1.p1" stex:srcref="paper.tex#textrange(from=75;0,to=87;19)">As the conversion to OMDoc is rather complex (the programs in the three steps take a
-variety of options), we support an automation by Unix Makefiles. There are three main
-<phrase class="code" xml:id="intro.p5.p2.p1.p1.p1" about="#intro.p5.p2.p1.p1.p1" stex:srcref="paper.tex#textrange(from=77;10,to=78;16)">make</phrase> targets.
-</p>
- </CMP>
- </omtext>
- <omgroup layout="description" xml:id="I2" about="#I2" stex:srcref="paper.tex#textrange(from=79;0,to=87;19)">
- <omtext xml:id="I2.p1" about="#I2.p1" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">
- <metadata xml:id="I2.p1.p1" about="#I2.p1.p1" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">
- <dc:title xml:id="I2.p1.p1.p1" about="#I2.p1.p1.p1" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">
- <p xml:id="I2.p1.p1.p1.p1" about="#I2.p1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">
- <phrase class="code" xml:id="I2.p1.p1.p1.p1.p1" about="#I2.p1.p1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">make omdoc</phrase>
- </p>
- </dc:title>
- </metadata>
- <CMP xml:id="I2.p1.p2" about="#I2.p1.p2" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">
- <p xml:id="I2.p1.p2.p1" about="#I2.p1.p2.p1" stex:srcref="paper.tex#textrange(from=79;53,to=80;31)">will trigger the OMDoc transformation of the target
-document.
-</p>
- </CMP>
- </omtext>
- <omtext xml:id="I2.p2" about="#I2.p2" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">
- <metadata xml:id="I2.p2.p1" about="#I2.p2.p1" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">
- <dc:title xml:id="I2.p2.p1.p1" about="#I2.p2.p1.p1" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">
- <p xml:id="I2.p2.p1.p1.p1" about="#I2.p2.p1.p1.p1" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">
- <phrase class="code" xml:id="I2.p2.p1.p1.p1.p1" about="#I2.p2.p1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">make mods</phrase>
- </p>
- </dc:title>
- </metadata>
- <CMP xml:id="I2.p2.p2" about="#I2.p2.p2" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">
- <p xml:id="I2.p2.p2.p1" about="#I2.p2.p2.p1" stex:srcref="paper.tex#textrange(from=81;55,to=82;31)">will trigger the OMDoc transformation of the modules.
-</p>
- </CMP>
- </omtext>
- <omtext xml:id="I2.p3" about="#I2.p3" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">
- <metadata xml:id="I2.p3.p1" about="#I2.p3.p1" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">
- <dc:title xml:id="I2.p3.p1.p1" about="#I2.p3.p1.p1" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">
- <p xml:id="I2.p3.p1.p1.p1" about="#I2.p3.p1.p1.p1" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">
- <phrase class="code" xml:id="I2.p3.p1.p1.p1.p1" about="#I2.p3.p1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">make pdf</phrase>
- </p>
- </dc:title>
- </metadata>
- <CMP xml:id="I2.p3.p2" about="#I2.p3.p2" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">
- <p xml:id="I2.p3.p2.p1" about="#I2.p3.p2.p1" stex:srcref="paper.tex#textrange(from=82;46,to=83;30)">will trigger the LaTeX formatting the target
-</p>
- </CMP>
- </omtext>
- <omtext xml:id="I2.p4" about="#I2.p4" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">
- <metadata xml:id="I2.p4.p1" about="#I2.p4.p1" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">
- <dc:title xml:id="I2.p4.p1.p1" about="#I2.p4.p1.p1" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">
- <p xml:id="I2.p4.p1.p1.p1" about="#I2.p4.p1.p1.p1" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">
- <phrase class="code" xml:id="I2.p4.p1.p1.p1.p1" about="#I2.p4.p1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">make mpdf</phrase>
- </p>
- </dc:title>
- </metadata>
- <CMP xml:id="I2.p4.p2" about="#I2.p4.p2" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">
- <p xml:id="I2.p4.p2.p1" about="#I2.p4.p2.p1" stex:srcref="paper.tex#textrange(from=83;47,to=84;31)">will trigger the LaTeX formatting the modules
-</p>
- </CMP>
- </omtext>
- <omtext xml:id="I2.p5" about="#I2.p5" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">
- <metadata xml:id="I2.p5.p1" about="#I2.p5.p1" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">
- <dc:title xml:id="I2.p5.p1.p1" about="#I2.p5.p1.p1" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">
- <p xml:id="I2.p5.p1.p1.p1" about="#I2.p5.p1.p1.p1" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">
- <phrase class="code" xml:id="I2.p5.p1.p1.p1.p1" about="#I2.p5.p1.p1.p1.p1" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">make sms</phrase>
- </p>
- </dc:title>
- </metadata>
- <CMP xml:id="I2.p5.p2" about="#I2.p5.p2" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">
- <p xml:id="I2.p5.p2.p1" about="#I2.p5.p2.p1" stex:srcref="paper.tex#textrange(from=84;57,to=85;30)">will trigger the re-generation of all semantic macro short
-forms of modules (this is implicitly called in all the other <phrase class="code" xml:id="I2.p5.p2.p1.p1" about="#I2.p5.p2.p1.p1" stex:srcref="paper.tex#textrange(from=85;10,to=86;79)">make</phrase> targets)
-</p>
- </CMP>
- </omtext>
- </omgroup>
- <omtext xml:id="intro.p5.p4" about="#intro.p5.p4" stex:srcref="paper.tex#textrange(from=75;0,to=108;13)">
- <CMP xml:id="intro.p5.p4.p1" about="#intro.p5.p4.p1" stex:srcref="paper.tex#textrange(from=75;0,to=108;13)">
- <p xml:id="intro.p5.p4.p1.p1" about="#intro.p5.p4.p1.p1" stex:srcref="paper.tex#textrange(from=75;0,to=75;42)">To use this, we need to set up a <phrase class="code" xml:id="intro.p5.p4.p1.p1.p1" about="#intro.p5.p4.p1.p1.p1" stex:srcref="paper.tex#textrange(from=87;24,to=88;55)">Makefile</phrase> of the following form:
-</p>
- <listingblock xmlns="http://dlmf.nist.gov/LaTeXML" class="language_make" xml:id="LSTx1">
- <tabular xml:id="LSTx1.p1" about="#LSTx1.p1">
- <tr xml:id="LSTx1.p1.p1" about="#LSTx1.p1.p1">
- <td class="linenumber" xml:id="LSTx1.p1.p1.p1" about="#LSTx1.p1.p1.p1"/>
- <td xml:id="LSTx1.p1.p1.p2" about="#LSTx1.p1.p1.p2">
- <text class="lstline">STEXDIR<text class="lstspace"> </text>=<text class="lstspace"> </text>../..</text>
- </td>
- </tr>
- <tr xml:id="LSTx1.p1.p2" about="#LSTx1.p1.p2">
- <td class="linenumber" xml:id="LSTx1.p1.p2.p1" about="#LSTx1.p1.p2.p1">2</td>
- <td xml:id="LSTx1.p1.p2.p2" about="#LSTx1.p1.p2.p2">
- <text class="lstline">TARGET<text class="lstspace"> </text>=<text class="lstspace"> </text>paper.tex</text>
- </td>
- </tr>
- <tr xml:id="LSTx1.p1.p3" about="#LSTx1.p1.p3">
- <td class="linenumber" xml:id="LSTx1.p1.p3.p1" about="#LSTx1.p1.p3.p1"/>
- <td xml:id="LSTx1.p1.p3.p2" about="#LSTx1.p1.p3.p2">
- <text class="lstline">MODSLIBDIR<text class="lstspace"> </text>=<text class="lstspace"> </text>../background</text>
- </td>
- </tr>
- <tr xml:id="LSTx1.p1.p4" about="#LSTx1.p1.p4">
- <td class="linenumber" xml:id="LSTx1.p1.p4.p1" about="#LSTx1.p1.p4.p1">4</td>
- <td xml:id="LSTx1.p1.p4.p2" about="#LSTx1.p1.p4.p2">
- <text class="lstline"/>
- </td>
- </tr>
- <tr xml:id="LSTx1.p1.p5" about="#LSTx1.p1.p5">
- <td class="linenumber" xml:id="LSTx1.p1.p5.p1" about="#LSTx1.p1.p5.p1"/>
- <td xml:id="LSTx1.p1.p5.p2" about="#LSTx1.p1.p5.p2">
- <text class="lstline">include<text class="lstspace"> </text>$(STEXDIR)/make/Makefile.vars</text>
- </td>
- </tr>
- <tr xml:id="LSTx1.p1.p6" about="#LSTx1.p1.p6">
- <td class="linenumber" xml:id="LSTx1.p1.p6.p1" about="#LSTx1.p1.p6.p1">6</td>
- <td xml:id="LSTx1.p1.p6.p2" about="#LSTx1.p1.p6.p2">
- <text class="lstline">all:<text class="lstspace"> </text>omdoc<text class="lstspace"> </text>mods<text class="lstspace"> </text>pdf</text>
- </td>
- </tr>
- <tr xml:id="LSTx1.p1.p7" about="#LSTx1.p1.p7">
- <td class="linenumber" xml:id="LSTx1.p1.p7.p1" about="#LSTx1.p1.p7.p1"/>
- <td xml:id="LSTx1.p1.p7.p2" about="#LSTx1.p1.p7.p2">
- <text class="lstline">include<text class="lstspace"> </text>$(STEXDIR)/make/Makefile.in</text>
- </td>
- </tr>
- </tabular>
- </listingblock>
- <p xml:id="intro.p5.p4.p1.p3" about="#intro.p5.p4.p1.p3" stex:srcref="paper.tex#textrange(from=75;0,to=75;42)">The variable <text xmlns="http://dlmf.nist.gov/LaTeXML" class="listing ">STEXDIR</text> has to be set to the main directory of the <phrase xml:id="intro.p5.p4.p1.p3.p1" about="#intro.p5.p4.p1.p3.p1" stex:srcref="paper.tex#textrange(from=92;1,to=93;86)">sTeX</phrase>
-distribution. The variable <text xmlns="http://dlmf.nist.gov/LaTeXML" class="listing ">TARGET</text> specifies the target document (all other
-<phrase class="code" xml:id="intro.p5.p4.p1.p3.p2" about="#intro.p5.p4.p1.p3.p2" stex:srcref="paper.tex#textrange(from=94;73,to=95;17)">*.tex</phrase> files that are not excluded in the <text xmlns="http://dlmf.nist.gov/LaTeXML" class="listing ">BUTFILES</text> variables are
-considered as modules). Here, the <phrase class="code" xml:id="intro.p5.p4.p1.p3.p3" about="#intro.p5.p4.p1.p3.p3" stex:srcref="paper.tex#textrange(from=95;32,to=96;56)">background</phrase> directory for convenience. The
-<text xmlns="http://dlmf.nist.gov/LaTeXML" class="listing ">MODSLIBDIR</text> specifies the location of the prefix and postfix files
-<phrase class="code" xml:id="intro.p5.p4.p1.p3.p4" about="#intro.p5.p4.p1.p3.p4" stex:srcref="paper.tex#textrange(from=97;68,to=98;19)">pre.tex</phrase> and <phrase class="code" xml:id="intro.p5.p4.p1.p3.p5" about="#intro.p5.p4.p1.p3.p5" stex:srcref="paper.tex#textrange(from=97;44,to=98;43)">post.tex</phrase> that will be prepended and appended to the
-modules to make them into complete files TeX files that can be converted. The last
-<!-- %**** paper.tex Line 100 **** -->three lines just include the Makefiles from the <phrase xml:id="intro.p5.p4.p1.p3.p6" about="#intro.p5.p4.p1.p3.p6" stex:srcref="paper.tex#textrange(from=99;32,to=100;56)">sTeX</phrase> distribution and configure the
-default make target (<phrase class="code" xml:id="intro.p5.p4.p1.p3.p7" about="#intro.p5.p4.p1.p3.p7" stex:srcref="paper.tex#textrange(from=100;31,to=101;40)">make all</phrase>) to be produce the pdf version
-</p>
- <p xml:id="intro.p5.p4.p1.p4" about="#intro.p5.p4.p1.p4" stex:srcref="paper.tex#textrange(from=75;0,to=108;13)">Note that in the directory <phrase class="code" xml:id="intro.p5.p4.p1.p4.p1" about="#intro.p5.p4.p1.p4.p1" stex:srcref="paper.tex#textrange(from=102;36,to=103;49)">background</phrase> we have a very similar Makefile as
-above. The only differences are that the variable <text xmlns="http://dlmf.nist.gov/LaTeXML" class="listing ">STEXDIR</text> is adapted and
-that the <text xmlns="http://dlmf.nist.gov/LaTeXML" class="listing ">BUTFILE</text> variable is set to <phrase class="code" xml:id="intro.p5.p4.p1.p4.p2" about="#intro.p5.p4.p1.p4.p2" stex:srcref="paper.tex#textrange(from=104;5,to=105;69)">pre.tex</phrase> and
-<phrase class="code" xml:id="intro.p5.p4.p1.p4.p3" about="#intro.p5.p4.p1.p4.p3" stex:srcref="paper.tex#textrange(from=105;51,to=106;20)">post.tex</phrase>, so that they are not converted. In the directory
-<phrase class="code" xml:id="intro.p5.p4.p1.p4.p4" about="#intro.p5.p4.p1.p4.p4" stex:srcref="paper.tex#textrange(from=106;56,to=107;22)">background</phrase> we have followed good practice by establishing a phony
-</p>
- </CMP>
- </omtext>
- </omgroup>
- </omgroup>
- <omgroup layout="sectioning" xml:id="sec.math" about="#sec.math" stex:srcref="paper.tex#textrange(from=112;0,to=116;13)">
- <metadata xml:id="sec.math.p1" about="#sec.math.p1" stex:srcref="paper.tex#textrange(from=112;0,to=112;50)">
- <dc:title stex:srcref="paper.tex#textrange(from=112;0,to=116;13)" xml:id="sec.math.p1.p1" about="#sec.math.p1.p1">
- <p xml:id="sec.math.p1.p1.p1" about="#sec.math.p1.p1.p1" stex:srcref="paper.tex#textrange(from=112;0,to=116;13)">Mathematical Content</p>
- </dc:title>
- </metadata>
- <omtext xml:id="sec.math.p2" about="#sec.math.p2" stex:srcref="paper.tex#textrange(from=112;0,to=13;13)">
- <CMP xml:id="sec.math.p2.p1" about="#sec.math.p2.p1" stex:srcref="paper.tex#textrange(from=112;0,to=13;13)">
- <p xml:id="sec.math.p2.p1.p1" about="#sec.math.p2.p1.p1" stex:srcref="paper.tex#textrange(from=112;0,to=13;13)">We present some standard mathematical definitions, here from calculus.
-</p>
- </CMP>
- </omtext>
- <omgroup layout="sectioning" xml:id="sec.contfuncs" about="#sec.contfuncs" stex:srcref="continuous.tex#textrange(from=1;0,to=13;13)">
- <metadata xml:id="sec.contfuncs.p1" about="#sec.contfuncs.p1" stex:srcref="continuous.tex#textrange(from=1;0,to=12;12)">
- <dc:title stex:srcref="continuous.tex#textrange(from=1;0,to=13;13)" xml:id="sec.contfuncs.p1.p1" about="#sec.contfuncs.p1.p1">
- <p xml:id="sec.contfuncs.p1.p1.p1" about="#sec.contfuncs.p1.p1.p1" stex:srcref="continuous.tex#textrange(from=1;0,to=13;13)">Continuous Functions</p>
- </dc:title>
- </metadata>
- <theory xml:id="continuous" about="#continuous" stex:srcref="continuous.tex#textrange(from=2;0,to=12;12)">
- <imports from="../background/functions.omdoc#functions" xml:id="continuous.p1" about="#continuous.p1" stex:srcref="continuous.tex#textrange(from=3;0,to=3;46)"/>
- <imports from="../background/reals.omdoc#reals" xml:id="continuous.p2" about="#continuous.p2" stex:srcref="continuous.tex#textrange(from=4;0,to=4;38)"/>
- <symbol name="continuousfunctions" xml:id="continuousfunctions.sym" about="#continuousfunctions.sym" stex:srcref="continuous.tex#textrange(from=5;1,to=5;54)"/>
- <notation cd="continuous" name="continuousfunctions">
- <prototype>
- <om:OMA>
- <om:OMS cd="continuous" cr="fun" name="continuousfunctions"/>
- <expr name="arg1"/>
- <expr name="arg2"/>
- </om:OMA>
- </prototype>
- <rendering>
- <Math xmlns="http://dlmf.nist.gov/LaTeXML" xml:id="continuous.m1" text="C ^ 0 * open-interval@(arg:1, arg:2)">
- <om:OMOBJ>
- <om:OMA>
- <om:OMS cd="arith1" name="times"/>
- <om:OMA>
- <om:OMS cd="ambiguous" name="superscript"/>
- <om:OMV name="C"/>
- <om:OMI>0</om:OMI>
- </om:OMA>
- <om:OMA>
- <om:OMS cd="latexml" name="open-interval"/>
- <om:OMV name="arg:1"/>
- <om:OMV name="arg:2"/>
- </om:OMA>
- </om:OMA>
- </om:OMOBJ>
- <m:math display="inline">
- <m:mrow>
- <m:msup>
- <m:mi mathvariant="script">C</m:mi>
- <m:mn>0</m:mn>
- </m:msup>
- <m:mo>⁢</m:mo>
- <m:mfenced open="(" close=")">
- <m:mrow>
- <m:mi>arg:1</m:mi>
- <m:mo>,</m:mo>
- <m:mi>arg:2</m:mi>
- </m:mrow>
- </m:mfenced>
- </m:mrow>
- </m:math>
- </Math>
- </rendering>
- </notation>
- <symbol name="continuous" xml:id="continuous.def.sym" about="#continuous.def.sym" stex:srcref="continuous.tex#textrange(from=7;1,to=11;16)"/>
- <definition for="continuous" xml:id="continuous.p5" about="#continuous.p5" stex:srcref="continuous.tex#textrange(from=7;1,to=11;16)">
- <CMP xml:id="continuous.p5.p1" about="#continuous.p5.p1" stex:srcref="continuous.tex#textrange(from=7;1,to=11;16)">
- <p xml:id="continuous.p5.p1.p1" about="#continuous.p5.p1.p1" stex:srcref="continuous.tex#textrange(from=7;1,to=11;16)">
-A function <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\fun{f}{\RealNumbers}{\RealNumbers}" xml:id="continuous.p5.p1.p1.m1" text="fun@(f, RealNumbers, RealNumbers)"><om:OMOBJ><om:OMA><om:OMS cd="functions" name="fun"/><om:OMV name="f"/><om:OMS cd="reals" name="RealNumbers"/><om:OMS cd="reals" name="RealNumbers"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">fun</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>f</m:mi><m:mo>, </m:mo><m:mi mathcolor="red">RealNumbers</m:mi><m:mo>, </m:mo><m:mi mathcolor="red">RealNumbers</m:mi></m:mrow></m:mrow></m:math></Math> is called <idx xml:id="continuous.p5.p1.p1.p2" about="#continuous.p5.p1.p1.p2" stex:srcref="continuous.tex#textrange(from=7;4,to=8;76)"><idt><term cd="continuous" name="continuous" role="definiendum" xml:id="continuous.p5.p1.p1.p2.p1" about="#continuous.p5.p1.p1.p2.p1" stex:srcref="continuous.tex#textrange(from=7;4,to=8;76)">continuous</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">continuous</text></idp></ide></idx> at
-<Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\inset{x}{\RealNumbers}" xml:id="continuous.p5.p1.p1.m2" text="inset@(x, RealNumbers)"><om:OMOBJ><om:OMA><om:OMS cd="functions" name="inset"/><om:OMV name="x"/><om:OMS cd="reals" name="RealNumbers"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">inset</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>x</m:mi><m:mo>, </m:mo><m:mi mathcolor="red">RealNumbers</m:mi></m:mrow></m:mrow></m:math></Math>, iff for all <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\epsilon&gt;0" xml:id="continuous.p5.p1.p1.m3" text="epsilon &gt; 0"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="greater-than"/><om:OMV name="ϵ"/><om:OMI>0</om:OMI></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi>ϵ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:mrow></m:math></Math> there is a <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\delta&gt;0" xml:id="continuous.p5.p1.p1.m4" text="delta &gt; 0"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="greater-than"/><om:OMV name="δ"/><om:OMI>0</om:OMI></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi>δ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:mrow></m:math></Math>, such that
-<Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\absval{f(x)-f(y)}&lt;\epsilon" xml:id="continuous.p5.p1.p1.m5" text="absval@(f * x - f * y) less epsilon"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="less-than"/><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="latexml" name="minus"/><om:OMA><om:OMS cd="arith1" name="times"/><om:OMV name="f"/><om:OMV name="x"/></om:OMA><om:OMA><om:OMS cd="arith1" name="times"/><om:OMV name="f"/><om:OMV name="y"/></om:OMA></om:OMA></om:OMA><om:OMV name="ϵ"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mrow><m:mi mathcolor="red">absval</m:mi><m:mo>⁡</m:mo><m:mrow><m:mrow><m:mi>f</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>x</m:mi></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>f</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>y</m:mi></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>&lt;</m:mo><m:mi>ϵ</m:mi></m:mrow></m:math></Math> for all <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\absval{x-y}&lt;\delta" xml:id="continuous.p5.p1.p1.m6" text="absval@(x - y) less delta"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="less-than"/><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="latexml" name="minus"/><om:OMV name="x"/><om:OMV name="y"/></om:OMA></om:OMA><om:OMV name="δ"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mrow><m:mi mathcolor="red">absval</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>y</m:mi></m:mrow></m:mrow><m:mo>&lt;</m:mo><m:mi>δ</m:mi></m:mrow></m:math></Math>
-</p>
- </CMP>
- </definition>
- </theory>
- </omgroup>
- <!-- %%% Local Variables:
- %%% mode: LaTeX
- %%% TeX-master: "paper"
- %%% End:-->
- <omgroup layout="sectioning" xml:id="sec.differentiable" about="#sec.differentiable" stex:srcref="differentiable.tex#textrange(from=1;0,to=12;13)">
- <metadata xml:id="sec.differentiable.p1" about="#sec.differentiable.p1" stex:srcref="differentiable.tex#textrange(from=1;0,to=11;12)">
- <dc:title stex:srcref="differentiable.tex#textrange(from=1;0,to=12;13)" xml:id="sec.differentiable.p1.p1" about="#sec.differentiable.p1.p1">
- <p xml:id="sec.differentiable.p1.p1.p1" about="#sec.differentiable.p1.p1.p1" stex:srcref="differentiable.tex#textrange(from=1;0,to=12;13)">Differentiable Functions</p>
- </dc:title>
- </metadata>
- <theory xml:id="differentiable" about="#differentiable" stex:srcref="differentiable.tex#textrange(from=2;0,to=11;12)">
- <imports from="./continuous.omdoc#continuous" xml:id="differentiable.p1" about="#differentiable.p1" stex:srcref="differentiable.tex#textrange(from=3;0,to=3;37)"/>
- <symbol name="difffunctions" xml:id="difffunctions.sym" about="#difffunctions.sym" stex:srcref="differentiable.tex#textrange(from=4;1,to=4;48)"/>
- <notation cd="differentiable" name="difffunctions">
- <prototype>
- <om:OMA>
- <om:OMS cd="differentiable" cr="fun" name="difffunctions"/>
- <expr name="arg1"/>
- <expr name="arg2"/>
- </om:OMA>
- </prototype>
- <rendering>
- <Math xmlns="http://dlmf.nist.gov/LaTeXML" xml:id="differentiable.m1" text="C ^ 1 * open-interval@(arg:1, arg:2)">
- <om:OMOBJ>
- <om:OMA>
- <om:OMS cd="arith1" name="times"/>
- <om:OMA>
- <om:OMS cd="ambiguous" name="superscript"/>
- <om:OMV name="C"/>
- <om:OMI>1</om:OMI>
- </om:OMA>
- <om:OMA>
- <om:OMS cd="latexml" name="open-interval"/>
- <om:OMV name="arg:1"/>
- <om:OMV name="arg:2"/>
- </om:OMA>
- </om:OMA>
- </om:OMOBJ>
- <m:math display="inline">
- <m:mrow>
- <m:msup>
- <m:mi mathvariant="script">C</m:mi>
- <m:mn>1</m:mn>
- </m:msup>
- <m:mo>⁢</m:mo>
- <m:mfenced open="(" close=")">
- <m:mrow>
- <m:mi>arg:1</m:mi>
- <m:mo>,</m:mo>
- <m:mi>arg:2</m:mi>
- </m:mrow>
- </m:mfenced>
- </m:mrow>
- </m:math>
- </Math>
- </rendering>
- </notation>
- <symbol name="differentiable" xml:id="differentiable.def.sym" about="#differentiable.def.sym" stex:srcref="differentiable.tex#textrange(from=6;1,to=10;16)"/>
- <definition for="differentiable continuous" xml:id="differentiable.p4" about="#differentiable.p4" stex:srcref="differentiable.tex#textrange(from=6;1,to=10;16)">
- <CMP xml:id="differentiable.p4.p1" about="#differentiable.p4.p1" stex:srcref="differentiable.tex#textrange(from=6;1,to=10;16)">
- <p xml:id="differentiable.p4.p1.p1" about="#differentiable.p4.p1.p1" stex:srcref="differentiable.tex#textrange(from=6;1,to=10;16)">
-A function <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\fun{f}{\RealNumbers}{\RealNumbers}" xml:id="differentiable.p4.p1.p1.m1" text="fun@(f, RealNumbers, RealNumbers)"><om:OMOBJ><om:OMA><om:OMS cd="functions" name="fun"/><om:OMV name="f"/><om:OMS cd="reals" name="RealNumbers"/><om:OMS cd="reals" name="RealNumbers"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">fun</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>f</m:mi><m:mo>, </m:mo><m:mi mathcolor="red">RealNumbers</m:mi><m:mo>, </m:mo><m:mi mathcolor="red">RealNumbers</m:mi></m:mrow></m:mrow></m:math></Math> is called <idx xml:id="differentiable.p4.p1.p1.p2" about="#differentiable.p4.p1.p1.p2" stex:srcref="differentiable.tex#textrange(from=6;4,to=7;80)"><idt><term cd="differentiable" name="differentiable" role="definiendum" xml:id="differentiable.p4.p1.p1.p2.p1" about="#differentiable.p4.p1.p1.p2.p1" stex:srcref="differentiable.tex#textrange(from=6;4,to=7;80)">differentiable</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">differentiable</text></idp></ide></idx> at
-<Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\inset{x}{\RealNumbers}" xml:id="differentiable.p4.p1.p1.m2" text="inset@(x, RealNumbers)"><om:OMOBJ><om:OMA><om:OMS cd="functions" name="inset"/><om:OMV name="x"/><om:OMS cd="reals" name="RealNumbers"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">inset</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>x</m:mi><m:mo>, </m:mo><m:mi mathcolor="red">RealNumbers</m:mi></m:mrow></m:mrow></m:math></Math>, iff for all <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\epsilon&gt;0" xml:id="differentiable.p4.p1.p1.m3" text="epsilon &gt; 0"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="greater-than"/><om:OMV name="ϵ"/><om:OMI>0</om:OMI></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi>ϵ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:mrow></m:math></Math> there is a <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\delta&gt;0" xml:id="differentiable.p4.p1.p1.m4" text="delta &gt; 0"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="greater-than"/><om:OMV name="δ"/><om:OMI>0</om:OMI></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi>δ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:mrow></m:math></Math>, such that
-<Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\rfrac{\absval{f(x)-f(y)}}{\absval{x-y}}&lt;\epsilon" xml:id="differentiable.p4.p1.p1.m5" text="rfrac@(absval@(f * x - f * y), absval@(x - y)) less epsilon"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="less-than"/><om:OMA><om:OMS cd="reals" name="rfrac"/><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="latexml" name="minus"/><om:OMA><om:OMS cd="arith1" name="times"/><om:OMV name="f"/><om:OMV name="x"/></om:OMA><om:OMA><om:OMS cd="arith1" name="times"/><om:OMV name="f"/><om:OMV name="y"/></om:OMA></om:OMA></om:OMA><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="latexml" name="minus"/><om:OMV name="x"/><om:OMV name="y"/></om:OMA></om:OMA></om:OMA><om:OMV name="ϵ"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mrow><m:mi mathcolor="red">rfrac</m:mi><m:mo>⁡</m:mo><m:mrow><m:mrow><m:mi mathcolor="red">absval</m:mi><m:mo>⁡</m:mo><m:mrow><m:mrow><m:mi>f</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>x</m:mi></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>f</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>y</m:mi></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>, </m:mo><m:mrow><m:mi mathcolor="red">absval</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>y</m:mi></m:mrow></m:mrow></m:mrow></m:mrow><m:mo>&lt;</m:mo><m:mi>ϵ</m:mi></m:mrow></m:math></Math> for all <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\absval{x-y}&lt;\delta" xml:id="differentiable.p4.p1.p1.m6" text="absval@(x - y) less delta"><om:OMOBJ><om:OMA><om:OMS cd="latexml" name="less-than"/><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="latexml" name="minus"/><om:OMV name="x"/><om:OMV name="y"/></om:OMA></om:OMA><om:OMV name="δ"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mrow><m:mi mathcolor="red">absval</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>y</m:mi></m:mrow></m:mrow><m:mo>&lt;</m:mo><m:mi>δ</m:mi></m:mrow></m:math></Math>.
-</p>
- </CMP>
- </definition>
- </theory>
- </omgroup>
- <!-- %%% Local Variables:
- %%% mode: LaTeX
- %%% TeX-master: "paper"
- %%% End:-->
- </omgroup>
- <omgroup layout="sectioning" xml:id="concl" about="#concl" stex:srcref="paper.tex#textrange(from=118;0,to=121;13)">
- <metadata xml:id="concl.p1" about="#concl.p1" stex:srcref="paper.tex#textrange(from=118;0,to=118;37)">
- <dc:title stex:srcref="paper.tex#textrange(from=118;0,to=121;13)" xml:id="concl.p1.p1" about="#concl.p1.p1">
- <p xml:id="concl.p1.p1.p1" about="#concl.p1.p1.p1" stex:srcref="paper.tex#textrange(from=118;0,to=121;13)">Conclusion</p>
- </dc:title>
- </metadata>
- <omtext xml:id="concl.p2" about="#concl.p2" stex:srcref="paper.tex#textrange(from=118;0,to=121;13)">
- <CMP xml:id="concl.p2.p1" about="#concl.p2.p1" stex:srcref="paper.tex#textrange(from=118;0,to=121;13)">
- <p xml:id="concl.p2.p1.p1" about="#concl.p2.p1.p1" stex:srcref="paper.tex#textrange(from=118;0,to=121;13)">In this note we have given an example of standard mathematical markup and shown how a a
-<phrase xml:id="concl.p2.p1.p1.p1" about="#concl.p2.p1.p1.p1" stex:srcref="paper.tex#textrange(from=119;42,to=120;8)">sTeX</phrase> collection can be set up for automation.
-</p>
- </CMP>
- </omtext>
- </omgroup>
- <bibliography xmlns="http://dlmf.nist.gov/LaTeXML" files="kwarc" xml:id="bib">
- <title>References</title>
- </bibliography>
-</omdoc>