summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/stex/example/paper/differentiable.xml
blob: 649b017e523cba7f082606834494be3dd312b7ed (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
<?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" 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 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:-->
</omdoc>