10.1 The Unit Step Function

A function is said to be discontinuous if it exhibits points of discontinuity, that is, if the function jumps from one value to another without taking on any intermediate values.

A well-known discontinuous function is the unit step function u ₀( t) ^[*] that is defined as

It is also represented by the waveform of Figure 10.1.

Figure 10.1: Waveform for u ₀( t)

In the waveform of Figure 10.1, the unit step function u ₀( t) changes abruptly from 0 to 1 at t = 0.

But if it changes at t = t ₀ instead, its waveform and definition are as shown in Figure 10.2.

Figure 10.2: Waveform and definition of u ₀( t ? t ₀)

Likewise, if the unit step function changes from 0 to 1 at t = ? t ₀ as shown in Figure 10.3, it is denoted as u ₀( t + t ₀)

Figure 10.3: Waveform and definition of u ₀( t + t ₀)

Other forms of the unit step function are...