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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"?>
<?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="Anonymous String#textrange(from=1;1,to=1;17)" xml:id="differentiable.omdoc">
  <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="post4"/>
                  <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:-->
</omdoc>