Answer

Verified

350.1k+ views

**Hint**: The solution for this question is in the descriptive way, we come across different kinds of discontinuities, but here in this question we have to explain or discuss the Jump discontinuity in detailed way and taking an example and plotting a graph for the example we can have a complete picture of a jump discontinuity.

**:**

__Complete step-by-step answer__Jump Discontinuity is a classification or type of discontinuities in which the function jumps, or steps, from one point to another along the curve of the function, often splitting the curve into two separate sections. While continuous functions are often used within mathematics, not all functions are continuous. The point on the domain of a function that is discontinuous is called the discontinuity.

A jump discontinuity occurs when the right-hand and left-hand limits exist but are not equal.

Consider an Example of a function with a jump discontinuity:

The discontinuous function \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}

{x + 1} \\

{ - x}

\end{array}\,\,\,\,\,\,\begin{array}{*{20}{c}}

{x > 0} \\

{x \geqslant 0}

\end{array}} \right.\]

For \[x > 0\],

\[ \Rightarrow \,\,\,\mathop {\lim }\limits_{x \to 0} f\left( x \right) = \mathop {\lim }\limits_{x \to 0} \left( {x + 1} \right)\]

\[ \Rightarrow \,\,\,\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 0 + 1\]

\[ \Rightarrow \,\,\,\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 1\]

But, for \[x = 0\]

\[ \Rightarrow \,\,f\left( 0 \right) = 0\] (Therefore, f is continuous from the left at 0, but not the right.)

Here, in this example in which \[\mathop {\lim }\limits_{x \to {0^ + }} \] exists, and \[\mathop {\lim }\limits_{x \to {0^ - }} \] also exists, but they are not equal.

The graphical representation of above example is:

Jump discontinuities are also called "discontinuities of the first kind". These kinds of discontinuities are big breaks in the graph, but not breaks at vertical asymptotes (those are specifically called infinite or essential discontinuities). You’ll often see jump discontinuities in piecewise-defined functions. A function is never continuous at a jump discontinuity, and it’s never differentiable there, either.

**Note**: In this question they have mentioned about the jump discontinuity, suppose if they given any other discontinuity like infinite discontinuity, removable discontinuity, end point discontinuity or mixed continuity these are all the types of discontinuities we have to explain each type of discontinuities, by giving an suitable example it’s better to explain or understand.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Establish a relation between electric current and drift class 12 physics CBSE

Guru Purnima speech in English in 100 words class 7 english CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Copper is not used as potentiometer wire because class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE