<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c9">g</font> =  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <font color="Maroon" title="c8">f</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1"/>
(  not  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1"/>
not  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">A</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c11">B</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">g</font> =  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c11">B</font></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topreala.html#T28" title="TOPREALA:th.28">TOPREALA:28</a></i>;<br/><a NAME="E5:29_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span> holds  <font color="Maroon" title="c13">h</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c15">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tmap_1.html#D2" title="TMAP_1:def.2">TMAP_1:def 2</a><br/>



<a NAME="E1:29_1_1_1_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">Z</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c8">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font><b> such that </b><br/><a NAME="E3:29_1_1_1_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c16">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">G</font>
 <b>by </b><i><a class="ref" href="topreala.html#T41" title="TOPREALA:th.41">TOPREALA:41</a></i>;<br/>
<a NAME="E4:29_1_1_1_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="ref" href="jordan16.html#D1" title="JORDAN16:def.1">JORDAN16:def 1</a></i>;<br/>
<a NAME="E5:29_1_1_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/>
<a NAME="E6:29_1_1_1_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:29_1_1_1_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="toprealb.html#E4:29_1_1_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c14">x</font><b> such that </b><br/><a NAME="E8:29_1_1_1_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c17">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Z</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:29_1_1_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="fcont_1.html#T5" title="FCONT_1:th.5">FCONT_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E10:29_1_1_1_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">g</font>
 <b>and </b><br/><a NAME="E11:29_1_1_1_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c17">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7" title="RCOMP_1:def.7">RCOMP_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">NN</font> = <font color="Maroon" title="c17">N</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E11:29_1_1_1_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="jordan6.html#T46" title="JORDAN6:th.46">JORDAN6:46</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">xx</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">M</font> = <font color="Maroon" title="c19">NN</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c10">A</font></span>)</span></span>)</span> ;<br/>
<a NAME="E15:29_1_1_1_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c19">NN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c19">NN</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E16:29_1_1_1_1_1"/>
( <font color="Maroon" title="c14">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0 &amp; <font color="Maroon" title="c14">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">g</font> )
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:29_1_1_1_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E17:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c19">NN</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:29_1_1_1_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="toprealb.html#E15:29_1_1_1_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E18:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">NN</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c20">xx</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">M</font> = <font color="Maroon" title="c21">M</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c14">x</font> <b>by </b><i><a class="ref" href="topreala.html#T26" title="TOPREALA:th.26">TOPREALA:26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c22">M</font>
;<br/>

<a NAME="E20:29_1_1_1_1_1"/>
<font color="Maroon" title="c22">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">NN</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E21:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c19">NN</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E22:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Z</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:29_1_1_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E23:29_1_1_1_1_1"/>
<font color="Maroon" title="c13">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">G</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:29_1_1_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:29_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1_1_1"/>
 <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <font color="Maroon" title="c8">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_2"/>
 <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <font color="Maroon" title="c8">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topreala.html#T27" title="TOPREALA:th.27">TOPREALA:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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