Step Function

Dhristi JEE 2022-24

Introduction to Step Function

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 step 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 

\[X_{A}(X) = \begin{cases}1; & ifx\varepsilon A,\\0; & ifx/\varepsilon A\end{cases}\]

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) = xx


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:

\[u(t) = \begin{cases}0 & t < 0\\1 & t > 0\end{cases}\]

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:


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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 the 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 functions 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 the integral of a step function, then the below formula is used for that case.

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

In the above picture, we can explain in 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 a slope of +1.

One such example of the integral of a step function is the 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 drawing a graph includes drawing a straight line for the outputs with their 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 a certain value is not included in that interval.


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

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

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


Tips to practice step function 

Step function being a maths topic is practical in nature. One can revise it by practising it thoroughly. Get hold of the previous year questions papers and mock tests. Pick out all the step function questions from these papers and mocks and attempt all the questions. This will enable the student to consolidate understanding of the above-mentioned knowledge provided in this article. It is advisable to practice in a timed manner sine it is adequate from the exams point of view. 


Focus areas of the topic 

Certain aspects of the step function topic require some special attention from the students. These are the areas that might potentially pose difficulty to students in the exams.  Focus on the graphs making aspect of the step function. 

 

It is one area where the students tend to make silly mistakes and get their answers wrong. Go through the above-mentioned methodology  To deal with the aspect of making graphs of this topic.

 

The students are also advised to practice integral functions well before going for the exams. Often it is seen, the complicated formula becomes hard to remember for the students. Therefore, practice it well.

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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.

3. What are the important tips to remember for making the graph?

In the topic step function, making a graph is one of the trickiest aspects. For making the graph the student shall begin with,   drawing a horizontal line segment to the constant output value over the constant of an input value that it corresponds to. Next, draw a circle point, at each included endpoint. At last,  draw an open circle at each horizontal line that is not included.  If the student follows the above mentioned instructions, making an error-free graph would be way easier for the students. 

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