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      <text>2.7.4.1 The unit step function</text>
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      <text><![CDATA[<h3>The unit step function (Heaviside step function)</h3>
<p>The unit step function is defined as follows:</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>A <b>polynomial expression</b> has the form:</p>
<p></p><p style="text-align: center;">\(u(t)=1\),&nbsp; &nbsp; for \(t \geq 0\),</p><p style="text-align: center;">\(u(t)=0\),&nbsp;&nbsp; for \(t&lt;0\).</p><p></p>
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<p>This is also known as the <b>Heaviside step function</b> in which case is is denote by \(H(x)\). Note that different conventions may be used at the jump point but we will use the above.</p>
<p>Study this definition carefully. You will see that it is defined in two parts, with one expression to be used when \(t\) is greater than or equal to 0, and another expression to be used when \(t\) is less than 0. The graph of this function is shown in the following figure. Note that the part of \(u(t)\) for which \(t&lt;0\) lies on the \(t\)-axis but, for clarity, is shown as a distinct dashed line.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/2.7.4.1%20Step%20function%201.svg" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="318" height="160">
</figure>
<p>There is a jump, or discontinuity in the graph when \(t= 0\). That is why we need to define the function in two parts; one part when \(t\) is negative, and one part when \(t\) is non-negative. The point with coordinates \((0,1)\) is part of the function defined on \(t≥0\).</p>
<p>The position of the discontinuity may be shifted to the left or right. The graph of \(u(t−d)\) is shown in the following picture:</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/2.7.4.2%20Shifted%20step%20function.svg" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="273" height="140">
</figure>
<p>In the previous two figures, the function takes the value 0 or 1.  We can adjust the value 1 by multiplying the function by any other number we choose. The graph of \(2u(t−3)\) is:</p><figure class="HELM_figure">
<img src="@@PLUGINFILE@@/2.7.4.3%20Step%20function3%20.svg" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="273" height="140">
</figure>]]></text>
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