In mathematics, a jump discontinuity occurs when a function experiences an abrupt change in value at a specific point in its domain. This type of discontinuity is characterized by the function having different left-hand and right-hand limits at the point of discontinuity, which do not equal each other. Essentially, the function “jumps” from one value to another, creating a distinct break in the graph.

In this article, we will discuss the concept of “Jump Discontinuty” including it’s definition, examples, as well as how to identify jump discontinuity.

What is Jump Discontinuity?

A jump discontinuity in mathematics refers to a type of discontinuity where the function exhibits a sudden change in value at a particular point in its domain. This occurs when the left-hand limit and the right-hand limit of the function at a specific point are both finite but are not equal to each other.

Definition Jump Discontinuity

For a function f(x) with a jump discontinuity at x = x₀, the following conditions hold:

• The left-hand limit [Tex] \lim_{{x \to x_0^-}} f(x) = L_1[/Tex]
• The right-hand limit [Tex]\lim_{{x \to x_0^+}} f(x) = L_2[/Tex]

L₁ and L₂ are finite numbers but L₁ ≠ L₂ .

Properties of Jump Discontinuity

Jump discontinuities have distinct characteristics that differentiate them from other types of discontinuities. Here are some key properties:

• At a jump discontinuity, the left-hand limit and the right-hand limit exist but are not equal.
• The values of the one-sided limits (L₁ and L₂) are finite. This distinguishes jump discontinuities from infinite discontinuities, where one or both limits are infinite
• On the graph of the function, a jump discontinuity is represented by a sudden break or gap between the two segments of the function.
• Jump discontinuities commonly occur in piecewise functions, where the function definition changes at specific points.
• Unlike removable discontinuities, which can be “fixed” by redefining the function at the discontinuity point, jump discontinuities cannot be removed simply by altering the function at a single point.
• A function with a jump discontinuity is not continuous at the point of discontinuity.

Examples of Jump Discontinuity

Some of the examples of jump discontinuity are:

Example 1: Piecewise Function

Consider the Heaviside step function H(x):

[Tex]f(x) = \begin{cases} 1 & \text{if } x < 2 \\ 3 & \text{if } x \geq 2 \end{cases} [/Tex]

At x = 2, there is a jump discontinuity as the function jumps from 1 to 3.

Example 2: Step Function (Heaviside Function)

Consider the function:

[Tex]H(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases} [/Tex]

At x = 0, the function jumps from 0 to 1, exhibiting a jump discontinuity.

Identifying Jump Discontinuity in Functions

There are two methods to identify jump discontinuity in any function i.e.

• Using Graphical Representation
• Using Algebraic Manipulation

Let’s discuss these methods in detail as follows:

Graphical Representation

This is because identifying jump discontinuities is most preferably done by use of graphical displays. With a jump discontinuity while plotting a function, two points on the graph will have a sharp transition up and down. This will be evident when drawing a graph since the graph most likely will indicate a break or a skip where the value of the function is instantly different.

Analytical Method

Analytically, to state that there is a jump discontinuity at a point x = c, thus arriving at the lateral limits of the function at x = c. If these limits exist and are different, the point should be to remove them and reach the maximum line. when there is a jump from one level of significance to completely different level, it is a jump discontinuity; in this case x = c. For a function f(x):

[Tex]\lim_{x \to c^-} f(x) = L_1[/Tex] and [Tex]\lim_{x \to c^+} f(x) = L_2 [/Tex] with L₁ ≠ L₂.

Conclusion

Jump discontinuities are integral components of calculus; it is necessary to have these to understand functions with the tendencies of a sudden change. In this article, the authors explain what jump discontinuity is, the features of the function with a jump discontinuity, some examples, and the importance of the study of this concept. Thus, through recognition and analysis of these gaps, students will be able to solve problems in different fields where such changes are present.

Read More,

• Continuity of Function
• Continuity and Discontinuity
• Peicewise Function

Solved Examples on Jump Discontinuity

Example 1: Find the points of jump discontinuity of the function:

[Tex]f(x) = \begin{cases} x + 2 & \text{if } x < 2 \\ 5 & \text{if } x = 2 \\ x – 1 & \text{if } x > 2 \end{cases}[/Tex]

Solution:

At x = 2,

Left-Hand Limit: lim_x→2− (x+2)=4

Right-Hand Limit: lim_x→2+ (x−1)=1

Since 4 ≠ 1, there is a jump discontinuity at x = 2.

Example 2: Determine if the function has a jump discontinuity at x=0:

[Tex]f(x) = \begin{cases} -1 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases}[/Tex]

​Solution:

At x = 0,

Left-Hand Limit: lim_x→0− f(x) = −1

Right-Hand Limit: lim_x→0+f(x) = 1

Since −1 ≠ 1, there is a jump discontinuity at x = 0.

Practice Problems on Jump Discontinuity

Problem 1: Determine if the function h(x) has a jump discontinuity at x = 3:

[Tex]h(x) = \begin{cases} x^2 & \text{if } x < 3 \\ 2x – 3 & \text{if } x \geq 3 \end{cases} [/Tex]

Problem 2: Identify the point of jump discontinuity for the function k(x):

[Tex]k(x) = \begin{cases} \sin(x) & \text{if } x < \pi \\ \cos(x) & \text{if } x \geq \pi \end{cases} [/Tex]

Problem 3: Verify if the function m(x) has a jump discontinuity at x = −2:

[Tex]k(x) = \begin{cases} \sin(x) & \text{if } x < \pi \\ \cos(x) & \text{if } x \geq \pi \end{cases} [/Tex]

Frequently Asked Questions – FAQs

What is a jump discontinuity?

A jump discontinuity is when the left-hand and/or right-hand limits of a function at a point exist and are not equal.

Can a function be continuous at a jump discontinuity?

This in turn means that a function cannot be continuous at a jump discontinuity since the limits as the variable approaches x from either side of the point of discontinuity are different.

How do you identify a jump discontinuity in a piecewise function?

To find a jump discontinuity, find the left and right handed derivatives at the point where the function is being investigated. If they are not equal, there is a jump discontinuity Since at x = t and t = a + t – a we get the same points which are not equal there is a jump discontinuity.

Is the value of the function at the point of jump discontinuity always defined?

No, the value of the function at the point of jump discontinuity may or may not be defined. If I may say, it can vary from the defined limits.

What is the practical significance of jump discontinuities?

Jump discontinuities are prominent in areas such as signal processing, engineering, and economics where there are sudden changes.

Can jump discontinuities occur in continuous functions?

No, continuous functions don’t have jump discontinuity because continuity of a function consists in its being possible fully to draw through the region bounded by the function and its axis without lifting the pencil.

How are jump discontinuities represented graphically?

From graphical point of view, it is depicted as a break or a vertical slash on the graph of the said function.