Discontinuity

In Maths, often there are functions f(x) that are not continuous at a point of its domain D. These non-continuous functions are called a point of discontinuity of the function. In other words, in a graph, if the functions are not connected to each other they will be called a discontinuous function.

The discontinuity can be because of the following situations: 

1. If both the right-hand limit as well as the left-hand limit or maybe any one of them do not exist.

2. If both of them i.e., the right-hand limit and the left-hand limit of a function do exist but are not equal. 

3. If either of the two or maybe both are not equal to the function f(x). 

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We can see that in the graph given above, the limits of the function towards the left and the limits of the function towards the right are unequal so the limit at x = 3 does not exist anymore. These types of functions are said to be a discontinuity of a function.

Types of Discontinuity

In this flow chart of the types of discontinuity, we can see that there are two types of discontinuity i.e., removable discontinuity and non-removable discontinuity. Removable discontinuity has two parts i.e., missing point and isolated point. Non-removable discontinuity has three parts i.e., finite type, infinite type, and oscillatory discontinuity. 

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What is a  Removable Discontinuity?

We can call a discontinuity “removable discontinuity” if the limit of the function exists but either they are not equal to the function or they are not defined. However, there is a possibility of redefining a function in a way that the limit will be equal to the value of the function at a particular point. 

Missing Point Discontinuity

A missing discontinuity arises when the limit of the function exists at a point but the function is undefined at that point. In a graph, it is represented as an open circle at the point where it is left undefined.  

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Isolated Point Discontinuity

In an isolated point discontinuity, the limits of a function do not only exist but are also defined at a point. This does not mean that they both are not equal. 

What is a Non - Removable Discontinuity?

If the limit of a function does not exist then we call it a non-removable discontinuity. It is not possible to redefine a function to make it continuous. A non-removable discontinuity can further be divided into 3 parts i.e., a finite type of a discontinuity, an infinite type of a discontinuity, and an oscillatory discontinuity. 

Finite Type 

In a finite type of discontinuity, both the left as well as the right-hand limits do exist but they are unequal. In other words, it is when a two-sided limit does not exist, but both the two one-sided limits are finite yet they are not equal to each other. In a graph of a finite type of a discontinuity, the function will be represented as a vertical gap between the two branches of the function. When there is a non-negative difference between the two limits, we call it the Jump of Discontinuity. 

Infinite Type

We can say it is an infinite discontinuity if either one or both the Right-Hand and the Left-Hand Limit do not exist or they are Infinite. We also call it Essential Discontinuity. If a graph of a function has the line x = k, as a vertical asymptote, then the function becomes either positively or negatively infinite. Therefore, the function f(x) will be called as an infinite discontinuity.

Oscillatory Discontinuity

The oscillatory discontinuity is a discontinuity when the limits oscillate between any two finite quantities.

Solved Examples

Question 1) Solve the discontinuity of a function algebraically and graph it. 

\[f(x) = \frac{(x - 2)(x + 2)(x - 1)}{(x - 1)}\]

Solution 1) We can remove or cancel the factor x = 1 from the numerator as well as the denominator. Therefore, we will be left with f(x) = (x - 2)(x + 1). since x = 1 is canceled, we get a removable discontinuity at x = 1. The graph will be represented as y = (x - 2)(x + 1) and a hole at x = 1. While graphing, y = (x - 2)(x + 1) as usual along with the hole.

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Question 2) Describe whether a function \[f(x) = sin\frac{1}{x}\] as continuity or discontinuity.

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Solution 2) The function is an oscillate infinitely as x approaches 0. The graph neither has a hole or a jump discontinuity nor does it shoot to infinity. However, it is also not continuous at x = 0.     

Question 3) Describe the discontinuity of the function from the graph given below:

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Solution 3) We can see that there are two jump discontinuity at x = -2 and x = 4. There is a removable discontinuity at x = 2 and an infinite discontinuity at x = 0.