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

<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> implies for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Olive">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E2:53_1"/>
for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1"/>
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Olive">x</font>
 ;<br/>

<b>thus </b><a NAME="E3:53"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a><br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/>



<a NAME="E1:53_2"/>
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:53_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div>
