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<li><a href="#rules-for-continuity" id="toc-rules-for-continuity" class="nav-link active" data-scroll-target="#rules-for-continuity"> <span class="header-section-number">20.1</span> Rules for continuity</a></li>
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<h1 class="title d-none d-lg-block"><span class="chapter-number">20</span>&nbsp; <span class="chapter-title">Continuity</span></h1>
</div>
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<p>This section uses these add-on packages:</p>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">CalculusWithJulia</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">SymPy</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<hr>
<p>The definition Google finds for <em>continuous</em> is <em>forming an unbroken whole; without interruption</em>.</p>
<p>The concept in calculus, as transferred to functions, is similar. Roughly speaking, a continuous function is one whose graph could be drawn without having to lift (or interrupt) the pencil drawing it.</p>
<p>Consider these two graphs:</p>
<div class="cell" data-hold="true" data-execution_count="4">
<div class="cell-output cell-output-display" data-execution_count="5">
<p><img src="continuity_files/figure-html/cell-5-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>and</p>
<div class="cell" data-hold="true" data-execution_count="5">
<div class="cell-output cell-output-display" data-execution_count="6">
<p><img src="continuity_files/figure-html/cell-6-output-1.svg" class="img-fluid"></p>
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<p>Though similar at some level - they agree at nearly every value of <span class="math inline">\(x\)</span> - the first has a “jump” from <span class="math inline">\(-1\)</span> to <span class="math inline">\(1\)</span> instead of the transition in the second one. The first is not continuous at <span class="math inline">\(0\)</span> - a break is needed to draw it - where as the second is continuous.</p>
<p>A formal definition of continuity was a bit harder to come about. At <a href="http://en.wikipedia.org/wiki/Intermediate_value_theorem">first</a> the concept was that for any <span class="math inline">\(y\)</span> between any two values in the range for <span class="math inline">\(f(x)\)</span>, the function should take on the value <span class="math inline">\(y\)</span> for some <span class="math inline">\(x\)</span>. Clearly this could distinguish the two graphs above, as one takes no values in <span class="math inline">\((-1,1)\)</span>, whereas the other - the continuous one - takes on all values in that range.</p>
<p>However, <a href="http://en.wikipedia.org/wiki/Cours_d%27Analyse">Cauchy</a> defined continuity by <span class="math inline">\(f(x + \alpha) - f(x)\)</span> being small whenever <span class="math inline">\(\alpha\)</span> was small. This basically rules out “jumps” and proves more useful as a tool to describe continuity.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Continuous_function#History">modern</a> definition simply pushes the details to the definition of the limit:</p>
<blockquote class="blockquote">
<p>A function <span class="math inline">\(f(x)\)</span> is continuous at <span class="math inline">\(x=c\)</span> if <span class="math inline">\(\lim_{x \rightarrow c}f(x) = f(c)\)</span>.</p>
</blockquote>
<p>This says three things</p>
<ul>
<li>The limit exists at <span class="math inline">\(c\)</span>.</li>
<li>The function is defined at <span class="math inline">\(c\)</span> (<span class="math inline">\(c\)</span> is in the domain).</li>
<li>The value of the limit is the same as <span class="math inline">\(f(c)\)</span>.</li>
</ul>
<p>This speaks to continuity at a point, we can extend this to continuity over an interval <span class="math inline">\((a,b)\)</span> by saying:</p>
<blockquote class="blockquote">
<p>A function <span class="math inline">\(f(x)\)</span> is continuous over <span class="math inline">\((a,b)\)</span> if at each point <span class="math inline">\(c\)</span> with <span class="math inline">\(a &lt; c &lt; b\)</span>, <span class="math inline">\(f(x)\)</span> is continuous at <span class="math inline">\(c\)</span>.</p>
</blockquote>
<p>Finally, as with limits, it can be convenient to speak of <em>right</em> continuity and <em>left</em> continuity at a point, where the limit in the defintion is replaced by a right or left limit, as appropriate.</p>
<div class="callout-warning callout callout-style-default callout-captioned">
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</div>
<div class="callout-caption-container flex-fill">
Warning
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</div>
<div class="callout-body-container callout-body">
<p>The limit in the definition of continuity is the basic limit and not an extended sense where infinities are accounted for.</p>
</div>
</div>
<section id="examples-of-continuity" class="level5">
<h5 class="anchored" data-anchor-id="examples-of-continuity">Examples of continuity</h5>
<p>Most familiar functions are continuous everywhere.</p>
<ul>
<li>For example, a monomial function <span class="math inline">\(f(x) = ax^n\)</span> for non-negative, integer <span class="math inline">\(n\)</span> will be continuous. This is because the limit exists everywhere, the domain of <span class="math inline">\(f\)</span> is all <span class="math inline">\(x\)</span> and there are no jumps.</li>
<li>Similarly, the basic trigonometric functions <span class="math inline">\(\sin(x)\)</span>, <span class="math inline">\(\cos(x)\)</span> are continuous everywhere.</li>
<li>So are the exponential functions <span class="math inline">\(f(x) = a^x, a &gt; 0\)</span>.</li>
<li>The hyperbolic sine (<span class="math inline">\((e^x - e^{-x})/2\)</span>) and cosine (<span class="math inline">\((e^x + e^{-x})/2\)</span>) are, as <span class="math inline">\(e^x\)</span> is.</li>
<li>The hyperbolic tangent is, as <span class="math inline">\(\cosh(x) &gt; 0\)</span> for all <span class="math inline">\(x\)</span>.</li>
</ul>
<p>Some familiar functions are <em>mostly</em> continuous but not everywhere.</p>
<ul>
<li>For example, <span class="math inline">\(f(x) = \sqrt{x}\)</span> is continuous on <span class="math inline">\((0,\infty)\)</span> and right continuous at <span class="math inline">\(0\)</span>, but it is not defined for negative <span class="math inline">\(x\)</span>, so can’t possibly be continuous there.</li>
<li>Similarly, <span class="math inline">\(f(x) = \log(x)\)</span> is continuous on <span class="math inline">\((0,\infty)\)</span>, but it is not defined at <span class="math inline">\(x=0\)</span>, so is not right continuous at <span class="math inline">\(0\)</span>.</li>
<li>The tangent function <span class="math inline">\(\tan(x) = \sin(x)/\cos(x)\)</span> is continuous everywhere <em>except</em> the points <span class="math inline">\(x\)</span> with <span class="math inline">\(\cos(x) = 0\)</span> (<span class="math inline">\(\pi/2 + k\pi, k\)</span> an integer).</li>
<li>The hyperbolic co-tangent is not continuous at <span class="math inline">\(x=0\)</span> – when <span class="math inline">\(\sinh\)</span> is <span class="math inline">\(0\)</span>,</li>
<li>The semicircle <span class="math inline">\(f(x) = \sqrt{1 - x^2}\)</span> is <em>continuous</em> on <span class="math inline">\((-1, 1)\)</span>. It is not continuous at <span class="math inline">\(-1\)</span> and <span class="math inline">\(1\)</span>, though it is right continuous at <span class="math inline">\(-1\)</span> and left continuous at <span class="math inline">\(1\)</span>.</li>
</ul>
</section>
<section id="examples-of-discontinuity" class="level5">
<h5 class="anchored" data-anchor-id="examples-of-discontinuity">Examples of discontinuity</h5>
<p>There are various reasons why a function may not be continuous.</p>
<ul>
<li>The function <span class="math inline">\(f(x) = \sin(x)/x\)</span> has a limit at <span class="math inline">\(0\)</span> but is not defined at <span class="math inline">\(0\)</span>, so is not continuous at <span class="math inline">\(0\)</span>. The function can be redefined to make it continuous.</li>
<li>The function <span class="math inline">\(f(x) = 1/x\)</span> is continuous everywhere <em>except</em> <span class="math inline">\(x=0\)</span> where <em>no</em> limit exists.</li>
<li>A rational function <span class="math inline">\(f(x) = p(x)/q(x)\)</span> will be continuous everywhere except where <span class="math inline">\(q(x)=0\)</span>. (The function <span class="math inline">\(f\)</span> may still have a limit where <span class="math inline">\(q\)</span> is <span class="math inline">\(0\)</span>, should factors cancel, but <span class="math inline">\(f\)</span> won’t be defined at such values.)</li>
<li>The function</li>
</ul>
<p><span class="math display">\[
f(x) = \begin{cases}
-1 &amp; x &lt; 0 \\
0 &amp; x = 0 \\
1 &amp; x &gt; 0
\end{cases}
\]</span></p>
<p>is implemented by <code>Julia</code>’s <code>sign</code> function. It has a value at <span class="math inline">\(0\)</span>, but no limit at <span class="math inline">\(0\)</span>, so is not continuous at <span class="math inline">\(0\)</span>. Furthermore, the left and right limits exist at <span class="math inline">\(0\)</span> but are not equal to <span class="math inline">\(f(0)\)</span> so the function is not left or right continuous at <span class="math inline">\(0\)</span>. It is continuous everywhere except at <span class="math inline">\(x=0\)</span>.</p>
<ul>
<li>Similarly, the function defined by this graph</li>
</ul>
<div class="cell" data-hold="true" data-execution_count="6">
<div class="cell-output cell-output-display" data-execution_count="7">
<p><img src="continuity_files/figure-html/cell-7-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>is not continuous at <span class="math inline">\(x=0\)</span>. It has a limit of <span class="math inline">\(0\)</span> at <span class="math inline">\(0\)</span>, a function value <span class="math inline">\(f(0) =1/2\)</span>, but the limit and the function value are not equal.</p>
<ul>
<li>The <code>floor</code> function, which rounds down to the nearest integer, is also not continuous at the integers, but is right continuous at the integers, as, for example, <span class="math inline">\(\lim_{x \rightarrow 0+} f(x) = f(0)\)</span>. This graph emphasizes the right continuity by placing a point for the value of the function when there is a jump:</li>
</ul>
<div class="cell" data-hold="true" data-execution_count="7">
<div class="cell-output cell-output-display" data-execution_count="8">
<p><img src="continuity_files/figure-html/cell-8-output-1.svg" class="img-fluid"></p>
</div>
</div>
<ul>
<li>The function <span class="math inline">\(f(x) = 1/x^2\)</span> is not continuous at <span class="math inline">\(x=0\)</span>: <span class="math inline">\(f(x)\)</span> is not defined at <span class="math inline">\(x=0\)</span> and <span class="math inline">\(f(x)\)</span> has no limit at <span class="math inline">\(x=0\)</span> (in the usual sense).</li>
<li>On the Wikipedia page for <a href="https://en.wikipedia.org/wiki/Continuous_function">continuity</a> the example of Dirichlet’s function is given:</li>
</ul>
<p><span class="math display">\[
f(x) =
\begin{cases}
0 &amp; \text{if } x \text{ is irrational,}\\
1 &amp; \text{if } x \text{ is rational.}
\end{cases}
\]</span></p>
<p>The limit for any <span class="math inline">\(c\)</span> is discontinuous, as any interval about <span class="math inline">\(c\)</span> will contain <em>both</em> rational and irrational numbers so the function will not take values in a small neighborhood around any potential <span class="math inline">\(L\)</span>.</p>
</section>
<section id="example" class="level5">
<h5 class="anchored" data-anchor-id="example">Example</h5>
<p>Let a function be defined by cases:</p>
<p><span class="math display">\[
f(x) = \begin{cases}
3x^2 + c &amp; x \geq 0,\\
2x-3 &amp; x &lt; 0.
\end{cases}
\]</span></p>
<p>What value of <span class="math inline">\(c\)</span> will make <span class="math inline">\(f(x)\)</span> a continuous function?</p>
<p>We note that for <span class="math inline">\(x &lt; 0\)</span> and for <span class="math inline">\(x &gt; 0\)</span> the function is a simple polynomial, so is continuous. At <span class="math inline">\(x=0\)</span> to be continuous we need a limit to exists and be equal to <span class="math inline">\(f(0)\)</span>, which is <span class="math inline">\(c\)</span>. A limit exists if the left and right limits are equal. This means we need to solve for <span class="math inline">\(c\)</span> to make the left and right limits equal. We do this next with a bit of overkill in this case:</p>
<div class="cell" data-execution_count="8">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> x c</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>ex1 <span class="op">=</span> <span class="fl">3</span>x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> c</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>ex2 <span class="op">=</span> <span class="fl">2</span>x<span class="op">-</span><span class="fl">3</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>del <span class="op">=</span> <span class="fu">limit</span>(ex1, x<span class="op">=&gt;</span><span class="fl">0</span>, dir<span class="op">=</span><span class="st">"+"</span>) <span class="op">-</span> <span class="fu">limit</span>(ex2, x<span class="op">=&gt;</span><span class="fl">0</span>, dir<span class="op">=</span><span class="st">"-"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="9">
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
\[
c + 3
\]
</span>
</div>
</div>
<p>We need to solve for <span class="math inline">\(c\)</span> to make <code>del</code> zero:</p>
<div class="cell" data-execution_count="9">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">solve</span>(del, c)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="10">
<pre><code>1-element Vector{Sym}:
-3</code></pre>
</div>
</div>
<p>This gives the value of <span class="math inline">\(c\)</span>.</p>
</section>
<section id="rules-for-continuity" class="level2" data-number="20.1">
<h2 data-number="20.1" class="anchored" data-anchor-id="rules-for-continuity"><span class="header-section-number">20.1</span> Rules for continuity</h2>
<p>As we’ve seen, functions can be combined in several ways. How do these relate with continuity?</p>
<p>Suppose <span class="math inline">\(f(x)\)</span> and <span class="math inline">\(g(x)\)</span> are both continuous on <span class="math inline">\(I\)</span>. Then</p>
<ul>
<li>The function <span class="math inline">\(h(x) = a f(x) + b g(x)\)</span> is continuous on <span class="math inline">\(I\)</span> for any real numbers <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>;</li>
<li>The function <span class="math inline">\(h(x) = f(x) \cdot g(x)\)</span> is continuous on <span class="math inline">\(I\)</span>; and</li>
<li>The function <span class="math inline">\(h(x) = f(x) / g(x)\)</span> is continuous at all points <span class="math inline">\(c\)</span> in <span class="math inline">\(I\)</span> <strong>where</strong> <span class="math inline">\(g(c) \neq 0\)</span>.</li>
<li>The function <span class="math inline">\(h(x) = f(g(x))\)</span> is continuous at <span class="math inline">\(x=c\)</span> <em>if</em> <span class="math inline">\(g(x)\)</span> is continuous at <span class="math inline">\(c\)</span> <em>and</em> <span class="math inline">\(f(x)\)</span> is continuous at <span class="math inline">\(g(c)\)</span>.</li>
</ul>
<p>So, continuity is preserved for all of the basic operations except when dividing by <span class="math inline">\(0\)</span>.</p>
<section id="examples" class="level5">
<h5 class="anchored" data-anchor-id="examples">Examples</h5>
<ul>
<li>Since a monomial <span class="math inline">\(f(x) = ax^n\)</span> (<span class="math inline">\(n\)</span> a non-negative integer) is continuous, by the first rule, any polynomial will be continuous.</li>
<li>Since both <span class="math inline">\(f(x) = e^x\)</span> and <span class="math inline">\(g(x)=\sin(x)\)</span> are continuous everywhere, so will be <span class="math inline">\(h(x) = e^x \cdot \sin(x)\)</span>.</li>
<li>Since <span class="math inline">\(f(x) = e^x\)</span> is continuous everywhere and <span class="math inline">\(g(x) = -x\)</span> is continuous everywhere, the composition <span class="math inline">\(h(x) = e^{-x}\)</span> will be continuous everywhere.</li>
<li>Since <span class="math inline">\(f(x) = x\)</span> is continuous everywhere, the function <span class="math inline">\(h(x) = 1/x\)</span> - a ratio of continuous functions - will be continuous everywhere <em>except</em> possibly at <span class="math inline">\(x=0\)</span> (where it is not continuous).</li>
<li>The function <span class="math inline">\(h(x) = e^{x\log(x)}\)</span> will be continuous on <span class="math inline">\((0,\infty)\)</span>, the same domain that <span class="math inline">\(g(x) = x\log(x)\)</span> is continuous. This function (also written as <span class="math inline">\(x^x\)</span>) has a right limit at <span class="math inline">\(0\)</span> (of <span class="math inline">\(1\)</span>), but is not right continuous, as <span class="math inline">\(h(0)\)</span> is not defined.</li>
</ul>
</section>
</section>
<section id="questions" class="level2" data-number="20.2">
<h2 data-number="20.2" class="anchored" data-anchor-id="questions"><span class="header-section-number">20.2</span> Questions</h2>
<section id="question" class="level6">
<h6 class="anchored" data-anchor-id="question">Question</h6>
<p>Let <span class="math inline">\(f(x) = \sin(x)\)</span> and <span class="math inline">\(g(x) = \cos(x)\)</span>. Which of these is not continuous everywhere?</p>
<p><span class="math display">\[
f+g,~ f-g,~ f\cdot g,~ f\circ g,~ f/g
\]</span></p>
<div class="cell" data-hold="true" data-execution_count="10">
<div class="cell-output cell-output-display" data-execution_count="11">
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<label class="form-check-label" for="radio_13574125576244883658_1">
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<span class="label-body px-1">
\(f\circ g\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_13574125576244883658_2">
<input class="form-check-input" type="radio" name="radio_13574125576244883658" id="radio_13574125576244883658_2" value="2">
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\(f\cdot g\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_13574125576244883658_3">
<input class="form-check-input" type="radio" name="radio_13574125576244883658" id="radio_13574125576244883658_3" value="3">
<span class="label-body px-1">
\(f+g\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_13574125576244883658_4">
<input class="form-check-input" type="radio" name="radio_13574125576244883658" id="radio_13574125576244883658_4" value="4">
<span class="label-body px-1">
\(f/g\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_13574125576244883658_5">
<input class="form-check-input" type="radio" name="radio_13574125576244883658" id="radio_13574125576244883658_5" value="5">
<span class="label-body px-1">
\(f-g\)
</span>
</label>
</div>
</div>
</div>
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</div>
</section>
<section id="question-1" class="level6">
<h6 class="anchored" data-anchor-id="question-1">Question</h6>
<p>Let <span class="math inline">\(f(x) = \sin(x)\)</span>, <span class="math inline">\(g(x) = \sqrt{x}\)</span>.</p>
<p>When will <span class="math inline">\(f\circ g\)</span> be continuous?</p>
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For all \(x\)
</span>
</label>
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<label class="form-check-label" for="radio_18322448394289550408_2">
<input class="form-check-input" type="radio" name="radio_18322448394289550408" id="radio_18322448394289550408_2" value="2">
<span class="label-body px-1">
For all \(x &gt; 0\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_18322448394289550408_3">
<input class="form-check-input" type="radio" name="radio_18322448394289550408" id="radio_18322448394289550408_3" value="3">
<span class="label-body px-1">
For all \(x\) where \(\sin(x) &gt; 0\)
</span>
</label>
</div>
</div>
</div>
<div id="18322448394289550408_message" style="padding-bottom: 15px"></div>
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</div>
<p>When will <span class="math inline">\(g \circ f\)</span> be continuous?</p>
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For all \(x\)
</span>
</label>
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<input class="form-check-input" type="radio" name="radio_2717208706857920862" id="radio_2717208706857920862_2" value="2">
<span class="label-body px-1">
For all \(x &gt; 0\)
</span>
</label>
</div>
<div class="form-check">
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<input class="form-check-input" type="radio" name="radio_2717208706857920862" id="radio_2717208706857920862_3" value="3">
<span class="label-body px-1">
For all \(x\) where \(\sin(x) &gt; 0\)
</span>
</label>
</div>
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</div>
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</section>
<section id="question-2" class="level6">
<h6 class="anchored" data-anchor-id="question-2">Question</h6>
<p>The composition <span class="math inline">\(f\circ g\)</span> will be continuous everywhere provided:</p>
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<span class="label-body px-1">
The function \(g\) is continuous everywhere
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_2153377839324383808_2">
<input class="form-check-input" type="radio" name="radio_2153377839324383808" id="radio_2153377839324383808_2" value="2">
<span class="label-body px-1">
The function \(f\) is continuous everywhere
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_2153377839324383808_3">
<input class="form-check-input" type="radio" name="radio_2153377839324383808" id="radio_2153377839324383808_3" value="3">
<span class="label-body px-1">
The function \(g\) is continuous everywhere and \(f\) is continuous on the range of \(g\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_2153377839324383808_4">
<input class="form-check-input" type="radio" name="radio_2153377839324383808" id="radio_2153377839324383808_4" value="4">
<span class="label-body px-1">
The function \(f\) is continuous everywhere and \(g\) is continuous on the range of \(f\)
</span>
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</section>
<section id="question-3" class="level6">
<h6 class="anchored" data-anchor-id="question-3">Question</h6>
<p>At which values is <span class="math inline">\(f(x) = 1/\sqrt{x-2}\)</span> not continuous?</p>
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For \(x \geq 0\)
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When \(x &gt; 2\)
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When \(x \geq 2\)
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When \(x \leq 2\)
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<section id="question-4" class="level6">
<h6 class="anchored" data-anchor-id="question-4">Question</h6>
<p>A value <span class="math inline">\(x=c\)</span> is a <em>removable singularity</em> for <span class="math inline">\(f(x)\)</span> if <span class="math inline">\(f(x)\)</span> is not continuous at <span class="math inline">\(c\)</span> but will be if <span class="math inline">\(f(c)\)</span> is redefined to be <span class="math inline">\(\lim_{x \rightarrow c} f(x)\)</span>.</p>
<p>The function <span class="math inline">\(f(x) = (x^2 - 4)/(x-2)\)</span> has a removable singularity at <span class="math inline">\(x=2\)</span>. What value would we redefine <span class="math inline">\(f(2)\)</span> to be, to make <span class="math inline">\(f\)</span> a continuous function?</p>
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<section id="question-5" class="level6">
<h6 class="anchored" data-anchor-id="question-5">Question</h6>
<p>The highly oscillatory function</p>
<p><span class="math display">\[
f(x) = x^2 (\cos(1/x) - 1)
\]</span></p>
<p>has a removable singularity at <span class="math inline">\(x=0\)</span>. What value would we redefine <span class="math inline">\(f(0)\)</span> to be, to make <span class="math inline">\(f\)</span> a continuous function?</p>
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</section>
<section id="question-6" class="level6">
<h6 class="anchored" data-anchor-id="question-6">Question</h6>
<p>Let <span class="math inline">\(f(x)\)</span> be defined by</p>
<p><span class="math display">\[
f(x) = \begin{cases}
c + \sin(2x - \pi/2) &amp; x &gt; 0\\
3x - 4 &amp; x \leq 0.
\end{cases}
\]</span></p>
<p>What value of <span class="math inline">\(c\)</span> will make <span class="math inline">\(f(x)\)</span> continuous?</p>
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</section>
<section id="question-7" class="level6">
<h6 class="anchored" data-anchor-id="question-7">Question</h6>
<p>Suppose <span class="math inline">\(f(x)\)</span>, <span class="math inline">\(g(x)\)</span>, and <span class="math inline">\(h(x)\)</span> are continuous functions on <span class="math inline">\((a,b)\)</span>. If <span class="math inline">\(a &lt; c &lt; b\)</span>, are you sure that <span class="math inline">\(lim_{x \rightarrow c} f(g(x))\)</span> is <span class="math inline">\(f(g(c))\)</span>?</p>
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No, as \(g(c)\) may not be in the interval \((a,b)\)
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Yes, composition of continuous functions results in a continuous function, so the limit is just the function value.
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<section id="question-8" class="level6">
<h6 class="anchored" data-anchor-id="question-8">Question</h6>
<p>Consider the function <span class="math inline">\(f(x)\)</span> given by the following graph</p>
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</div>
<p>The function <span class="math inline">\(f(x)\)</span> is continuous at <span class="math inline">\(x=1\)</span>?</p>
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Yes
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<p>The function <span class="math inline">\(f(x)\)</span> is continuous at <span class="math inline">\(x=2\)</span>?</p>
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No
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<p>The function <span class="math inline">\(f(x)\)</span> is right continuous at <span class="math inline">\(x=3\)</span>?</p>
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Yes
</span>
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<p>The function <span class="math inline">\(f(x)\)</span> is left continuous at <span class="math inline">\(x=4\)</span>?</p>
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<section id="question-9" class="level6">
<h6 class="anchored" data-anchor-id="question-9">Question</h6>
<p>Let <span class="math inline">\(f(x)\)</span> and <span class="math inline">\(g(x)\)</span> be continuous functions whose graph of <span class="math inline">\([0,1]\)</span> is given by:</p>
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<p><img src="continuity_files/figure-html/cell-25-output-1.svg" class="img-fluid"></p>
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</div>
<p>What is <span class="math inline">\(\lim_{x \rightarrow 0.25} f(g(x))\)</span>?</p>
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<p>What is <span class="math inline">\(\lim{x \rightarrow 0.25} g(f(x))\)</span>?</p>
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<p>What is <span class="math inline">\(\lim_{x \rightarrow 0.5} f(g(x))\)</span>?</p>
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