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      <text>2.4.1.3 Continuous and discontinuous functions and limits (cont.)</text>
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      <text><![CDATA[<p>When defining a discontinuous function algebraically it is often necessary to give different function rules for different values of <span class="math inline">\(x\)</span>. Consider, for example, the function defined as:</p>
\[f(x)= \begin{cases}
3 &amp; x &lt; 0 \\
x^2 &amp; x \geq 0
\end{cases}\]
<p>Notice that there is one rule for when <span class="math inline">\(x\)</span> is less than 0 and another rule for when <span class="math inline">\(x\)</span> is greater than or equal to 0.</p><p>A graph of this function is shown in the figure below.</p><p style="text-align: center;"><figure class="HELM_figure">
<img src="@@PLUGINFILE@@/block6-fig2-1.svg" alt="A discontinuous function" class="img-responsive atto_image_button_text-bottom" width="220" height="191">
<figcaption>An example of a discontinuous function</figcaption>
</figure>
<p>Suppose we ask ‘to what value does <span class="math inline">\(y\)</span> approach as <span class="math inline">\(x\)</span> approaches 0?’. From the graph we see that as <span class="math inline">\(x\)</span> gets nearer and nearer to 0, the value of <span class="math inline">\(y\)</span> gets nearer to 0, if we approach from the right-hand side. We write this formally as</p><p><span class="math inline">\(\qquad \lim_{x\rightarrow 0^+} f(x) = 0\)</span></p><p>and say ‘the limit of <span class="math inline">\(f(x)\)</span> as <span class="math inline">\(x\)</span> tends to 0 from above is 0.’</p><p>On the other hand if <span class="math inline">\(x\)</span> gets closer to zero, from the left-hand side, the value of <span class="math inline">\(y\)</span> remains at 3. In this case we write</p><p><span class="math inline">\(\qquad \lim_{x\rightarrow 0^-}f(x) = 3\)</span></p><p>and say ‘the limit of <span class="math inline">\(f(x)\)</span> as <span class="math inline">\(x\)</span> tends to 0 from below is 3.’</p><p>In this example the right-hand limit and the left-hand limit are not equal, and this is indicative of the fact that the function is discontinuous.</p>
<p>In general a function is continuous at a point <span class="math inline">\(x=a\)</span> if the left-hand and right-hand limits are the same there and are finite, and if both of these are equal to the value of the function at that point. That is</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>A function \(f(x)\) is continuous at \(x=a\) if and only if: <span class="math display">\[\qquad \lim_{x\rightarrow a^+} f(x) = \lim_{x\rightarrow a^-} f(x) = f(a)\]</span></p>
</div>
<p>If the right-hand and left-hand limits are the same, we can simply describe this common limit as <span class="math inline">\(\lim_{x\rightarrow a} f(x)\)</span>. If the limits are not the same we say the limit of the function does not exist at <span class="math inline">\(x=a\)</span>.</p>]]></text>
<file name="block6-fig2-1.svg" path="/" encoding="base64">[encoded data, 9768 characters]</file>
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