# Discontinuity

Think of a rubber band that you can stretch. When you pull it slowly from both ends, it gets longer and longer. But if you pull it too hard, it might break!

This concept is similar to Discontinuity in mathematics. In this article, we will explore what Discontinuity means and understand it better!

## Introduction to Discontinuity

Discontinuity helps us understand how some equations or functions behave unpredictably. Similar to how a rubber band can only stretch to a certain limit, discontinuity shows us the limits of an equation.

**Also Check: Continuity and Discontinuity**

## Definition of Discontinuity

Discontinuity is an important concept in mathematics, especially in calculus, analysis, and topology. It refers to a point, function, or sequence that is not continuous at a certain value or interval. Simply put, it's where a function, sequence, or curve doesn't flow smoothly.

**Common cases of discontinuity include:**

- When the left and right-hand limits exist but are not equal. This means the function approaches different values from the left and right, causing a sudden change at that point.
- When the left and right-hand limits are equal to each other but not to the function's value at that point. This creates a hole or gap in the function.
- When either the left or right-hand limit (or both) does not exist, breaking the smooth flow of the function.

A function with breaks or interruptions in its graph is called a discontinuous function.

If the left and right-hand limits both exist but are unequal, it's called a first-kind discontinuity. If only the left-hand limit exists but is unequal to the function's value, it's a first-kind discontinuity from the left.

**Also Check: Difference Between Variance and Standard Deviation**

## Types of Discontinuity

There are several types of discontinuity, including:

- Jump Discontinuity
- Infinite Discontinuity
- Removable Discontinuity

### Jump Discontinuity

A jump discontinuity occurs when a function's value abruptly changes at a specific point. For example, a function might smoothly progress but then suddenly jump to a different value.

Examples:

- |3.14| = 3
- |-2.7| = -3

Suppose at *x = n*, where *n* is an integer, the function jumps to *n + 1*, causing a discontinuity.

Jump discontinuities can be:

**Discontinuity of the First Kind:**The left and right-hand limits exist but are unequal.**Discontinuity of the Second Kind:**Neither the left nor right-hand limits exist.

**Also Check: Cosine Function**

### Removable Discontinuity

A removable discontinuity occurs when the left and right-hand limits of a function are unequal to the function's value at a certain point. This happens when there is a gap or hole in the function, which can be "fixed" by redefining the function at that point.

Example:

f(x) = 1/(x - 2). Here, *x = 2* causes a hole. The function can be made continuous by defining *f(2)*.

### Infinite Discontinuity

Infinite discontinuity, or essential discontinuity, happens when either one or both the right and left-hand limits do not exist or are infinite. This occurs when the function approaches infinity as *x* approaches a certain value.

Example:

f(x) = 1/(x - 2). At *x = 2*, the function becomes infinitely large (positive or negative), indicating an infinite discontinuity.

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## Frequently Asked Questions on Discontinuity

A discontinuity is a point where a function is not continuous. It is a break or gap in the graph of a function, where the function is not defined or has a sudden jump in value.

To check if a function has a discontinuity at a point, you need to evaluate the left-hand limit, right-hand limit, and the function value at that point. If either limit does not exist or is not equal to the function value, then the function has a discontinuity at that point.

Discontinuities can be classified into three main types: jump discontinuities, removable discontinuities, and infinite discontinuities. Jump discontinuities occur when the left and right limits exist but are not equal, removable discontinuities happen when the limits exist and are equal but not equal to the function value, and infinite discontinuities arise when at least one limit is infinite.

To identify discontinuities, you can plot the graph of the function and look for breaks or jumps in the graph. Alternatively, you can evaluate the limits from the left and right sides of a point and compare them to the function value at that point. If the limits do not exist or are not equal to the function value, then the function has a discontinuity at that point.

The three main types of discontinuity are:

- Jump discontinuities: where the left and right limits exist but are not equal
- Removable discontinuities: where the limits exist and are equal but not equal to the function value
- Infinite discontinuities: where at least one limit is infinite