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<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><b>let </b><font color="Maroon" title="c4">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/>

<a NAME="E3:51"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  holds  <font color="Maroon" title="c3">f</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="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<a NAME="E1:51_1"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:51_1"/>
<font color="Maroon" title="c5">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:51_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="topreala.html#E2:51"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E4:51_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="topreala.html#E1:51_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1"/>
<font color="Maroon" title="c3">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="topreala.html#E1:51"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="topreala.html#T42" target="_self" title="TOPREALA:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
<font color="Maroon" title="c3">f</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/>


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