# Heaviside Function: Clarifying H(x-a) & H(a-x)

• I
• Sudhir Regmi
In summary, the conversation revolves around the Heaviside step function and whether H(x-a) and H(a-x) are the same or different. While initially it was thought that H(x)=-H(-x), it is later clarified that the correct definition of the function is H(x) = 0 for x<0 and H(x) = 1 for x>0. Therefore, H(x-a) and H(a-x) are not the same, as shown by their respective graphs.
Sudhir Regmi
Hi everyone,
I could not figure out whether H(x-a) and H(a-x) are same or different. Please help me to understand this.

Member warned to be careful of posting incorrect information
Since H(x)=-H(-x), it is true.
Hope this helps.

Replusz said:
Since H(x)=-H(-x), it is true.
Hope this helps.
Thanks, it means H(a-x) = -H(x-a) is true.

Exactly :)

Replusz said:
Since H(x)=-H(-x), it is true.
Hope this helps.
The Heaviside step function is usually define: H(x) = 0 for x<0, and H(x) = 1 for x>0. So it is not true that H(x) = -H(-x). For example, H(1) = 1 but H(-1) = 0.

Jason

No. The Heaviside function is H(0)=0 and H(x<0)= -1 and H(x>1)=+1

Replusz said:
No. The Heaviside function is H(0)=0 and H(x<0)= -1 and H(x>1)=+1
Replusz said:
Take another look at that graph near the top of the page that you linked to. If x < 0, H(x) = 0, not -1.

Mark44 said:
Take another look at that graph near the top of the page that you linked to. If x < 0, H(x) = 0, not -1.
Hello Mark44,
What do you think about my original question, is H( x-a) equal to H( a-x)? Where x is a variable and a is a constant.

Sudhir Regmi said:
Hello Mark44,
What do you think about my original question, is H( x-a) equal to H( a-x)? Where x is a variable and a is a constant.
##H(x - a) = \begin{cases} 1 & x > a \\ 0 & x < a \\ \end{cases}##
H(a - x) = H(-(x - a)) -- this is the reflection of the graph of H(x - a) across the line x = a, so the graphs of these two functions are not the same.

Thank you Mark44, Replusz and jasonRF for responding to my question.

## 1. What is the Heaviside function?

The Heaviside function, also known as the unit step function, is a mathematical function that is defined as 0 for negative values and 1 for positive values. It is commonly denoted as H(x).

## 2. What does H(x-a) represent?

H(x-a) represents a shifted version of the Heaviside function, where the value of a determines the location of the step. For x < a, H(x-a) will have a value of 0, and for x > a, it will have a value of 1.

## 3. How is the Heaviside function used in mathematics?

The Heaviside function is commonly used in mathematics to model discontinuous functions and to define piecewise functions. It is also used in solving differential equations and in signal processing.

## 4. What is the relationship between H(x-a) and H(a-x)?

H(x-a) and H(a-x) are mirror images of each other. H(x-a) will have a step at x = a, while H(a-x) will have a step at x = -a. In other words, the two functions are reflections across the y-axis.

## 5. How is the Heaviside function related to the Dirac delta function?

The Heaviside function is the derivative of the Dirac delta function. This means that the Heaviside function can be used to represent the integral of the Dirac delta function. Additionally, the Dirac delta function can be defined using the Heaviside function, as δ(x) = dH(x)/dx.

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