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+<?xml version="1.0"?>
+<!--This OMDoc document is generated from an sTeX-encoded one via LaTeXML, you may want to reconsider editing it.-->
+<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="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 xmlns:mcd="http://www.w3.org/ns/mathml-cd" xml:id="differentiable" about="#differentiable" stex:srcref="differentiable.tex#textrange(from=2;0,to=11;12)">
+ <imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/>
+ <imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/>
+ <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>
+ <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>
+ <render name="arg1"/>
+ <m:mo>,</m:mo>
+ <render name="arg2"/>
+ </m:mrow>
+ </m:mfenced>
+ </m:mrow>
+ </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 <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> 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><phrase>differentiable</phrase></idp></ide></idx> at
+<om:OMOBJ><om:OMA><om:OMS cd="functions" name="inset"/><om:OMV name="x"/><om:OMS cd="reals" name="RealNumbers"/></om:OMA></om:OMOBJ>, iff for all <om:OMOBJ><om:OMA><om:OMS cd="relation1" name="gt"/><om:OMV name="&#x3F5;"/><om:OMI>0</om:OMI></om:OMA></om:OMOBJ> there is a <om:OMOBJ><om:OMA><om:OMS cd="relation1" name="gt"/><om:OMV name="&#x3B4;"/><om:OMI>0</om:OMI></om:OMA></om:OMOBJ>, such that
+<om:OMOBJ><om:OMA><om:OMS cd="relation1" name="lt"/><om:OMA><om:OMS cd="reals" name="rfrac"/><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="arith1" 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="arith1" name="minus"/><om:OMV name="x"/><om:OMV name="y"/></om:OMA></om:OMA></om:OMA><om:OMV name="&#x3F5;"/></om:OMA></om:OMOBJ> for all <om:OMOBJ><om:OMA><om:OMS cd="relation1" name="lt"/><om:OMA><om:OMS cd="reals" name="absval"/><om:OMA><om:OMS cd="arith1" name="minus"/><om:OMV name="x"/><om:OMV name="y"/></om:OMA></om:OMA><om:OMV name="&#x3B4;"/></om:OMA></om:OMOBJ>.
+</p>
+ </CMP>
+ </definition>
+ </theory>
+ </omgroup>
+ <!--The data elements-->
+</omdoc>