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254 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Continuous Functions</h1>
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<p>A function is continuous when its graph is a single unbroken curve ...</p>
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<div class="center80"><img src="../images/style/pencil-paper.svg" alt="pencil" style="float:left; margin: 0 10px 5px 0;" width="119" height="99" >
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<p class="center"><span class="large">... that you could draw without lifting your pen from the paper</span>.</p>
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</div>
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<p>That is not a formal definition, but it helps you understand the idea.</p>
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<p>Here is a continuous function:</p>
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<p class="center"><img src="images/continuous-polynomial.svg" alt="continuous polynomial" width="198" height="198" ></p>
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<h2>Examples</h2>
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<p>So what is <b>not continuous</b> (also called <b>discontinuous</b>) ?</p>
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<p>Look out for holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).</p>
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<p><br></p>
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<div class="flexy">
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<div class="col">
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<img src="images/continuous-hole-no.svg" alt="not continuous hole " width="135" height="102" ><br>
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<span class="larger"><b>Not</b> Continuous</span><br>
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(hole)
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</div>
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<div class="col">
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<img src="images/continuous-jump-no.svg" alt="not continuous jump " width="135" height="102" ><br>
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<span class="larger"><b>Not</b> Continuous</span><br>
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(jump)
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</div>
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<div class="col">
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<img src="images/continuous-asymp-no.svg" alt="not continuous asymptote " width="135" height="102" ><br>
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<span class="larger"><b>Not</b> Continuous</span><br>
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(vertical asymptote)
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</div>
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</div>
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<p><br></p>
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<p>Try these different functions so you get the idea:</p>
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<script>
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continuityexamplesMain();
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</script>
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<p class="center">(Use slider to zoom, drag graph to reposition, click graph to re-center.)</p>
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</noscript>
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</div>
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-->
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<h2>Domain</h2>
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<p style="float:right; margin: 0 0 5px 25px;"><img src="../sets/images/range-domain-graph.svg" alt="doman and range" width="298" height="167" ></p>
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<p>A function has a <a href="../sets/domain-range-codomain.html">Domain.</a></p>
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<p>In its simplest form the domain is all the values that <b>go into</b> a function.</p>
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<div style="clear:both"></div>
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<p class="center large">We may be able to choose a domain that makes the function continuous</p>
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<p> </p>
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<div class="example">
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<h3>Example: 1/(x−1)</h3>
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<p>At x=1 we have:</p>
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<div class="so"> 1/(1−1) = 1/0 = undefined<br>
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</div>
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<p>So there is a "discontinuity" at x=1</p>
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<p class="center"><img src="images/continuous-asymp-include.svg" alt="continuous asymptote" width="191" height="146" ><span class="larger"><br>
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f(x) = 1/(x−1)</span></p>
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<p><span class="larger">So f(x) = 1/(x−1) over <b>all Real Numbers</b> is NOT continuous</span></p>
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<p> </p>
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<p>Let's change the domain to <b>x>1</b></p>
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<p class="center"><img src="images/continuous-asymp-exclude.svg" alt="continuous asymptote exclude" width="191" height="146" ><br>
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g(x) = 1/(x−1) for <b>x>1</b></p>
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<p><span class="larger">So g(x) IS continuous</span></p>
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<p> </p>
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<p>In other words g(x) does <b>not</b> include the value x=1, so it is <b>continuous</b>.</p>
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</div>
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<p>When a function is <b>continuous within its Domain</b>, it is a continuous function.</p>
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<h2>More Formally !</h2>
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<p>We can define <b>continuous</b> using <a href="limits.html">Limits</a> (it helps to read that page first):</p>
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<div class="def">
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<p class="large">A function <b>f</b> is continuous when, for <b>every</b> value <b>c</b> in its Domain:</p>
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<p class="center large">f(c) is defined,</p>
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<p class="center">and</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>x→c</strong></span><i>f(x) = f(c)</i></p>
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<p class="center"><i>"the limit of f(x) as x approaches c equals f(c)</i>"</p>
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</div>
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<p><span class="center">The limit says: </span></p>
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<p class="center larger">"as x gets closer and closer to c<br>
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then f(x) gets closer and closer to f(c)"</p>
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<p>And we have to check from both directions:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">as x approaches c (from left)<br>
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then f(x) approaches f(c)</td>
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<td> </td>
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<td><span class="center"><img src="images/continuous-limit-graph.gif" alt="continuous limit graph" width="285" height="202" ></span></td>
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</tr>
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<tr style="text-align:center;">
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<td style="text-align:right;"> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td style="text-align:right;">AND<br>
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as x approaches c (from right)<br>
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then f(x) approaches f(c)</td>
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<td> </td>
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<td><span class="center"><img src="images/continuous-limit-graph2.gif" alt="continuous limit graph" width="283" height="203" ></span></td>
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</tr>
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</tbody></table>
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<p>If we get different values from left and right (a "jump"), then the limit does not exist!</p>
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<p>And remember this has to be true for every value <b>c</b> in the domain.</p>
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<h2>How to Use:</h2>
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<p>Make sure that, for all <b>x</b> values:</p>
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<ul>
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<li><b> f(x)</b> is defined</li>
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<li>and the limit at <b>x</b> equals <b>f(x)</b></li>
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</ul>
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<p>Here are some examples:</p>
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<div class="example">
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<h3>Example: f(x) = (x<sup>2</sup>−1)/(x−1) for all Real Numbers</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-x2-1-x-1.svg" alt="graph (x^2-1)/(x-1) hole" width="137" height="147" ></p>
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<p>The function is <b>undefined</b> when x=1:</p>
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<p class="center">(x<sup>2</sup>−1)/(x−1) = (1<sup>2</sup>−1)/(1−1) = <b>0/0</b></p>
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<p>So it is <b>not</b> a continuous function</p>
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</div>
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<p>Let us change the domain:</p>
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<div class="example">
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<h3>Example: g(x) = (x<sup>2</sup>−1)/(x−1) over the interval x<1</h3>
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<p><b>Almost</b> the same function, but now it is over an interval that does <b>not</b> include x=1.</p>
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<p>So now it <b>is</b> a continuous function (does not include the "hole")</p>
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</div>
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<div class="example">
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<h3>Example: How about this <a href="../sets/functions-piecewise.html">piecewise function</a>:</h3>
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<p class="center"><img src="images/continuous-jump-fn.svg" alt="h(x) = { 2 if x<=1, x if x>1 }" width="249" height="81" ></p>
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<p>It looks like this:</p>
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<p class="center"><img src="images/continuous-jump-graph.gif" alt="continuous jump graph" width="178" height="159" ></p>
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<p>It is <b>defined</b> at x=1, because <b>h(1)=2</b> (no "hole")</p>
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<p>But at x=1 <b>you can't say what the limit is</b>, because there are two competing answers:</p>
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<ul>
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<li>"2" from the left, and</li>
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<li>"1" from the right</li>
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</ul>
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<p>so in fact the limit does not exist at x=1 (there is a "jump")</p>
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<p>And so the function is <b>not continuous</b>.</p>
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</div>
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<p>But:</p>
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<div class="example">
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<h3>Example: How about the piecewise function <a href="../sets/function-absolute-value.html">absolute value</a>:</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../sets/images/function-absolute-b.svg" alt="Absolute Value function" width="339" height="81" ></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../sets/images/function-absolute.svg" alt="Absolute Value function" width="241" height="241" ></p>
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<div style="clear:both"></div>
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<p>At x=0 it has a very pointy change!</p>
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<p>But it is still <b>defined</b> at x=0, because <b>f(0)=0</b> (so no "hole"),</p>
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<p>And the limit as you approach x=0 (from either side) is also <b>0</b> (so no "jump"),</p>
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<p>So it is in fact <b> continuous</b>.</p>
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<p>(But it is not <a href="differentiable.html">differentiable</a> at x=0)</p>
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</div>
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<p> </p>
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<div class="related">
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<a href="differentiable.html">Differentiable</a>
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<a href="index.html">Calculus Index</a>
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