Step Functions

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Section 8.3 Step Functions

Definition 8.3.1. The Step or Heaviside Function.

The step function \(\Theta(x)\text{,}\) also called the Heaviside function or theta function, is defined to be \(0\) if \(x\lt 0\) and \(1\) if \(x>0\text{.}\) See Figure 8.3.2. A function that behaves differently in two or more different regions is called piecewise and often written

\begin{equation} \Theta(x)=\begin{cases} 0 \amp x\lt 0\\ \frac{1}{2} \amp x=0\\ 1 \amp x\gt 0 \end{cases}\tag{8.3.1} \end{equation}

Figure 8.3.2. The step function \(\Theta(x)\text{.}\)

In most physical problems, it is unnecessary to know the exact value of \(\Theta(x)\) at the discontinuity. If it ever matters, it is usually easiest to define it symmetrically, i.e.

\begin{equation} \Theta(0) = \frac12 .\tag{8.3.2} \end{equation}

Step functions are used to model idealized physical situations where some quantity changes rapidly from one value to another in such a way that the exact details of the change are irrelevant for the solution of the problem, e.g. edges of materials or a process that switches on abruptly at a particular time, etc.

By shifting the argument of the function, it is possible to put the discontinuity of the theta function wherever we need it. In Figure 8.3.3 you can see that the graph of \(\Theta(x-2)\) has discontinuity at \(x=2\text{,}\) instead of at \(x=0\text{.}\)

Figure 8.3.3. The function \(\Theta (x-2)\text{.}\)

Most often the step function appears in contexts where it is multiplying some other function \(f(x)\text{.}\) In this case, the step function can be thought of as a switch, that ``turns on’’ the function \(f(x)\text{.}\) You can watch the operation of the switch in Figure 8.3.4 below.

Figure 8.3.4. An applet that allows you to change the value of \(\delta\) in \(f(x)\, \Theta(x-\delta)\text{.}\) The default function displayed is \(f(x)=\cos(x)\text{.}\)

The step function can also be used to turn a function off, since

\begin{equation} \Theta(-x) = 1-\Theta(x)\tag{8.3.3} \end{equation}

is a step down, rather than up.

Question 8.3.1. Try It Yourself: Graphing Step Functions.

Make sure that you are able to graph all of the following functions:

\begin{align*} f(x) \amp = 2\Theta(x) ,\\ f(x) \amp = \Theta(x-2) ,\\ f(x) \amp = \Theta(2x-3) ,\\ f(x) \amp = \Theta(2-x) ,\\ f(x) \amp = \Theta(x)\> \Theta(x-2) ,\\ f(x) \amp = \Theta(x)-\Theta(x-2) ,\\ f(x) \amp = \sin x\> \Theta(x-\frac{\pi}{2}) . \end{align*}