Step Function

Unit Step Function Definition

Mathematics is all about functions and equations to solve a given problem. The step function is one such type. As the name suggests, a step function is sometimes called the staircase function. We can also define it as the constant function on the real numbers. It is a piecewise constant function on the finite set. The other name for the set function is floor function or the greatest integer function which is the combination of the linear functions for a defined interval. However, in the case of mathematics, the definition for this function is entirely different.

Basic Definition of The Step Function

To define a step function given by f: R→R, which is discontinuous, you can write it in the form of:

f(x) = \[\sum_{i=0}^{n}\alpha _{i}x_{Ai}\] (x)

In the above equation, x is defined for the real numbers. 

α is the real number and A is defined for the interval with the condition n >= 0. 

If this condition is satisfied, then the indicator function A will be given by X

XA Is given by 

xA (x) = \[\left\{\begin{matrix} 1;if x \epsilon A,\\ 0;if x \not\epsilon A \end{matrix}\right.\]

The value of the function is 1 if x belongs to A and the value is 0 if x does not belong to A.

Alternatively, a function given by f: R→R is called the greatest integer function when x belongs to a real number and y = f(x) = [x]

What is Unit Step?

A unit step function is also defined as a Heaviside function. In this, the value of the given function keeps changing after the given time interval given by t. We generally define unit step function by u(t) which is further denoted by the below unit step signal:

u(t) = \[\left\{\begin{matrix} 0 & t< 0\\ 1& t>0 \end{matrix}\right.\]

In the above formula, u is defined as the function of time. If time goes negative, the value of u(t) is 0. However, if time is positive, the value is 1. 

Considering the graph for the above equation, it will be given by:

(Image to be added soon)

The graph of the unit step function clearly satisfies the above equation. It is the graph of 

f(t) = u(t)

Derivative of Step Function

The function works for all the levels except for the case of t =0. Hence the derivative of step function becomes zero for all values of t. However, it becomes infinite when t = 0. 

In the unit step function, its derivative is known as an impulse function. Engineers use impulse function to draw a model for certain events. However, the value of impulse function is zero for most of the cases. 

Integral of Step Function

If you want to compute integral of a step function, then the below formula is used for that case. 

\[\int_{-\infty}^{t}u(s)ds\] = \[\begin{Bmatrix} 0, & if t < 0 \\ \int_{0}^{t}ds=t, & if t\geq 0 \end{Bmatrix}\] = tu(t)

In the above picture, we can explain it in the simple words that a unit step function is the ramp function. If the value is less than 0 for t, then all values are zero. When t = 0, the function defines the straight line to infinity. The straight line is formed with the slope of +1.

One such example of integral of a step function is Piecewise function. However, it forms a definite integral.

Unit Step Function Graph

For a step function, drawing a graph is similar to graphing a piecewise function. However, in this, we will draw a graph for each step separately. To draw the graph for a unit step function, you need to follow certain steps as given below:

  • The very first step to draw a graph includes drawing a straight line for the outputs with its corresponding inputs. 

  • If for a line drawn, you want to include the value of the endpoints, then represent it using a filled circle. 

  • Third and the final step is to draw an empty or an open circle on the second endpoint of the horizontal line. Open circle indicates that the certain value is not included in that interval.

(Image to be added soon)

The above diagram is the description of the points mentioned to keep in mind before drawing a graph for step function. 

Properties of Step Function

Below are the mentioned properties of the step function:

  • When we add or multiply two-step functions, then also it gives step function as the output. 

  • When you multiply a step function with a number, it will give a step function. Thus for real numbers, steps functions are responsible for creating algebra. 

  • In a step function, there is only a finite number of values as output. 

FAQs (Frequently Asked Questions)

1. Is a step function continuous?

A step function is similar to the staircase function, which consists of the constant pieces in it. After the definite interval, these pieces are gone through some observation. However, there could be a change in the value of function after the definite time interval. Thus the changing value after a certain time interval states that a step function is discontinuous. In the case of a unit step function, if the time is discrete, then the unit step is given by a perfect sequence. However, if the time interval is continuous and not discrete, then there is a complication of discontinuity in the unit step.

2. Why do we use a step function?

If you want to work on session-based applications, then step function is the right method, to begin with. To check the coordination of the different steps and the smooth functioning of an eCommerce website, step function plays an important role. The value for a step function keeps changing abruptly after a certain time interval given by t. One such example of the use of a step function is changing voltage,  turning on or off after some time. Thus we have a complete description for switching with its mathematical equation. The primary use of a step function is in the engineering applications.