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

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="toprealb.html#K6" target="_self">R^1</a> <font color="Maroon">c<sub>8</sub></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">dom</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> or  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">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">dom</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="toprealb.html#K6" target="_self">R^1</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topreala.html#T28">TOPREALA:28</a></i>;<br/><a NAME="E5:29_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> holds <font color="Maroon">c<sub>13</sub></font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Olive">b<sub>1</sub></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">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a><br/>



<i><font color="Green">E19</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> = 
 <a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>

;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E3:29_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="topreala.html#T41">TOPREALA:41</a></i>;<br/>
<a NAME="E4:29_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="jordan16.html#D1">JORDAN16:def 1</a></i>;<br/>
<a NAME="E5:29_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<a NAME="E6:29_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:29_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="toprealb.html#E4:29_1_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E8:29_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:29_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fcont_1.html#T5">FCONT_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E10:29_1_1_1_1_1"/><i><font color="Green">E24</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E11:29_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E11:29_1_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan6.html#T46">JORDAN6:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E15:29_1_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E16:29_1_1_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K6">-</a> 0 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:29_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E17:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:29_1_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="toprealb.html#E15:29_1_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E18:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="ref" href="topreala.html#T26">TOPREALA:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<a NAME="E20:29_1_1_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E21:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E22:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:29_1_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E23:29_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:29_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T1">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">c<sub>13</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49">TMAP_1:49</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1_1_1"/>
 <a href="toprealb.html#K6" target="_self">R^1</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T8">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">dom</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E3:29_1_1_1_2"/><b>then </b>
 <a href="toprealb.html#K6" target="_self">R^1</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_1_2"/>
 <a href="toprealb.html#K6" target="_self">R^1</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topreala.html#T27">TOPREALA:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
