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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" 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>
                <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>
                <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>
                <td xml:id="LSTx1.p1.p5.p2" about="#LSTx1.p1.p5.p2">
                  <text class="lstline">include<text class="lstspace"> </text>$(STEXDIR)/make/Makefile.vars</text>
                </td>
              </tr>
              <tr xml:id="LSTx1.p1.p6" about="#LSTx1.p1.p6">
                <td class="linenumber" xml:id="LSTx1.p1.p6.p1" about="#LSTx1.p1.p6.p1">6</td>
                <td xml:id="LSTx1.p1.p6.p2" about="#LSTx1.p1.p6.p2">
                  <text class="lstline">all:<text class="lstspace"> </text>omdoc<text class="lstspace"> </text>mods<text class="lstspace"> </text>pdf</text>
                </td>
              </tr>
              <tr xml:id="LSTx1.p1.p7" about="#LSTx1.p1.p7">
                <td class="linenumber" xml:id="LSTx1.p1.p7.p1" about="#LSTx1.p1.p7.p1"></td>
                <td xml:id="LSTx1.p1.p7.p2" about="#LSTx1.p1.p7.p2">
                  <text class="lstline">include<text class="lstspace"> </text>$(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)">
              <XMath>
                <XMApp>
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMApp>
                    <XMTok role="SUPERSCRIPTOP" scriptpos="post6"/>
                    <XMTok role="UNKNOWN" possibleFunction="yes" font="caligraphic">C</XMTok>
                    <XMTok meaning="0" role="NUMBER">0</XMTok>
                  </XMApp>
                  <XMApp>
                    <XMTok meaning="open-interval" role="FENCED" argclose=")" argopen="(" separators=","/>
                    <XMTok role="ID">arg:1</XMTok>
                    <XMTok role="ID">arg:2</XMTok>
                  </XMApp>
                </XMApp>
              </XMath>
            </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)"><XMath><XMApp><XMTok meaning="fun" name="fun" omcd="functions"/><XMTok role="UNKNOWN" font="italic">f</XMTok><XMTok meaning="RealNumbers" name="RealNumbers" omcd="reals"/><XMTok meaning="RealNumbers" name="RealNumbers" omcd="reals"/></XMApp></XMath></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)"><XMath><XMApp><XMTok meaning="inset" name="inset" omcd="functions"/><XMTok role="UNKNOWN" font="italic">x</XMTok><XMTok meaning="RealNumbers" name="RealNumbers" omcd="reals"/></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="greater-than" role="RELOP">&gt;</XMTok><XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok><XMTok meaning="0" role="NUMBER">0</XMTok></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="greater-than" role="RELOP">&gt;</XMTok><XMTok name="delta" role="UNKNOWN" font="italic">δ</XMTok><XMTok meaning="0" role="NUMBER">0</XMTok></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="less-than" role="RELOP">&lt;</XMTok><XMApp><XMTok meaning="absval" name="absval" omcd="reals"/><XMApp><XMTok meaning="minus" role="ADDOP">-</XMTok><XMApp><XMTok meaning="times" role="MULOP">⁢</XMTok><XMTok role="UNKNOWN" possibleFunction="yes" font="italic">f</XMTok><XMTok role="UNKNOWN" open="(" close=")" font="italic">x</XMTok></XMApp><XMApp><XMTok meaning="times" role="MULOP">⁢</XMTok><XMTok role="UNKNOWN" possibleFunction="yes" font="italic">f</XMTok><XMTok role="UNKNOWN" open="(" close=")" font="italic">y</XMTok></XMApp></XMApp></XMApp><XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="less-than" role="RELOP">&lt;</XMTok><XMApp><XMTok meaning="absval" name="absval" omcd="reals"/><XMApp><XMTok meaning="minus" role="ADDOP">-</XMTok><XMTok role="UNKNOWN" font="italic">x</XMTok><XMTok role="UNKNOWN" font="italic">y</XMTok></XMApp></XMApp><XMTok name="delta" role="UNKNOWN" font="italic">δ</XMTok></XMApp></XMath></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)">
              <XMath>
                <XMApp>
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMApp>
                    <XMTok role="SUPERSCRIPTOP" scriptpos="post6"/>
                    <XMTok role="UNKNOWN" possibleFunction="yes" font="caligraphic">C</XMTok>
                    <XMTok meaning="1" role="NUMBER">1</XMTok>
                  </XMApp>
                  <XMApp>
                    <XMTok meaning="open-interval" role="FENCED" argclose=")" argopen="(" separators=","/>
                    <XMTok role="ID">arg:1</XMTok>
                    <XMTok role="ID">arg:2</XMTok>
                  </XMApp>
                </XMApp>
              </XMath>
            </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)"><XMath><XMApp><XMTok meaning="fun" name="fun" omcd="functions"/><XMTok role="UNKNOWN" font="italic">f</XMTok><XMTok meaning="RealNumbers" name="RealNumbers" omcd="reals"/><XMTok meaning="RealNumbers" name="RealNumbers" omcd="reals"/></XMApp></XMath></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)"><XMath><XMApp><XMTok meaning="inset" name="inset" omcd="functions"/><XMTok role="UNKNOWN" font="italic">x</XMTok><XMTok meaning="RealNumbers" name="RealNumbers" omcd="reals"/></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="greater-than" role="RELOP">&gt;</XMTok><XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok><XMTok meaning="0" role="NUMBER">0</XMTok></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="greater-than" role="RELOP">&gt;</XMTok><XMTok name="delta" role="UNKNOWN" font="italic">δ</XMTok><XMTok meaning="0" role="NUMBER">0</XMTok></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="less-than" role="RELOP">&lt;</XMTok><XMApp><XMTok meaning="rfrac" name="rfrac" omcd="reals"/><XMApp><XMTok meaning="absval" name="absval" omcd="reals"/><XMApp><XMTok meaning="minus" role="ADDOP">-</XMTok><XMApp><XMTok meaning="times" role="MULOP">⁢</XMTok><XMTok role="UNKNOWN" possibleFunction="yes" font="italic">f</XMTok><XMTok role="UNKNOWN" open="(" close=")" font="italic">x</XMTok></XMApp><XMApp><XMTok meaning="times" role="MULOP">⁢</XMTok><XMTok role="UNKNOWN" possibleFunction="yes" font="italic">f</XMTok><XMTok role="UNKNOWN" open="(" close=")" font="italic">y</XMTok></XMApp></XMApp></XMApp><XMApp><XMTok meaning="absval" name="absval" omcd="reals"/><XMApp><XMTok meaning="minus" role="ADDOP">-</XMTok><XMTok role="UNKNOWN" font="italic">x</XMTok><XMTok role="UNKNOWN" font="italic">y</XMTok></XMApp></XMApp></XMApp><XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok></XMApp></XMath></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"><XMath><XMApp><XMTok meaning="less-than" role="RELOP">&lt;</XMTok><XMApp><XMTok meaning="absval" name="absval" omcd="reals"/><XMApp><XMTok meaning="minus" role="ADDOP">-</XMTok><XMTok role="UNKNOWN" font="italic">x</XMTok><XMTok role="UNKNOWN" font="italic">y</XMTok></XMApp></XMApp><XMTok name="delta" role="UNKNOWN" font="italic">δ</XMTok></XMApp></XMath></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>