Discontinuity 

In Mathematics, discontinuity refers to a point at which a function is not continuous. A function is continuous at a point if its left-hand limit, right-hand limit, and the function’s value at that point are all equal. If any of these conditions fail, the function is said to have a discontinuity there. Discontinuity plays a crucial role in calculus, helping to analyze function behavior, identify asymptotes, and solve advanced problems in limits, differentiation, and integration.

1.0What Is Discontinuity of a Function?

A function f(x) is said to be discontinuous at point x = a if any of the following holds:

• The left-hand limit or right-hand limit does not exist.
• Both limits exist but are unequal.
• The limits exist, but the function’s value f(a) does not match the limits.

2.0Discontinuity in Piecewise Functions

Piecewise functions often exhibit discontinuities where the expression changes from one part to another.

Example: 

f(x)= $\{\begin{array}{ll}{x}^2,& x\le 2\\ 5x-4,& x>2\end{array}$

At x = 2:

• ${\lim }_{x\to 2^{-}}f(x)=(2{)}^2=4$
• ${\lim }_{x\to 2^{+}}f(x)=5(2)-4=6$

Since limits are not equal, f(x) is discontinuous at x = 2.

3.0How to Find Discontinuity of a Function?

1. Check for points where the function is undefined.
2. Calculate left-hand and right-hand limits.
3. Compare limits with the function’s value at the point.
4. Classify the type of discontinuity (Jump, Removable, Infinite).

4.0Why Is Discontinuity Important?

• Helps in understanding function behavior in calculus.
• Important for graph sketching and limit evaluations.
• Crucial in solving integration and differentiation problems.
• Relevant for JEE Advanced and other competitive exams.

5.0How Many Types of Discontinuity Are There?

In mathematics, there are three main types of discontinuity in a function:

1. Jump Discontinuity
2. Removable Discontinuity
3. Infinite (Essential) Discontinuity

Let’s discuss each of these types in detail along with relevant examples.

Jump Discontinuity

A jump discontinuity occurs when the left-hand limit and right-hand limit of a function at a point both exist and are finite, but they are not equal. The function "jumps" from one value to another at that point.

• Mathematical Condition:

${\lim }_{x\to a^{-}}f(x)\ne {\lim }_{x\to a^{+}}f(x)$

Example – Jump Discontinuity 

f(x)=$\{\begin{array}{ll}2x+1,& x<1\\ 3x-2,& x\ge 1\end{array}$

At x = 1:

• Left-hand limit: 3
• Right-hand limit: 1

Since they are unequal, this is a Jump Discontinuity.

Removable Discontinuity

A removable discontinuity occurs when the left and right limits exist and are equal, but the function is either undefined at that point or differs from the limit.

• Mathematical Condition:

${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=L\text{but}f(a)\ne L\text{or}f(a)\text{ undefined}$

Example – Removable Discontinuity

$f(x)=\frac{x^2-1}{x-1}$

Simplifies to f(x) = x + 1 for $x\ne 1$ but f(1) is undefined.

${\lim }_{x\to 1}f(x)=2$

Thus, there is a Removable Discontinuity at x = 1.

Infinite (Essential) Discontinuity

An infinite discontinuity occurs when the function approaches infinity near the point of discontinuity.

• Mathematical Condition:

${\lim }_{x\to a^{-}}f(x)=\pm \infty \text{or}{\lim }_{x\to a^{+}}f(x)=\pm \infty$

Example – Infinite Discontinuity

$f(x)=\frac{1}{x-2}$

At x = 2:

• Left-hand limit: -$\infty$
• Right-hand limit: $\infty$

This is an Infinite Discontinuity.

6.0Solved Examples on Discontinuity

Example 1 – Jump Discontinuity

Consider the function:  

f(x)=$\{\begin{array}{ll}x+2,& x<1\\ 2x+1,& x\ge 1\end{array}$

Determine whether there is a discontinuity at x = 1, and if yes, identify the type.

Solution:

• Left-hand limit:

${\lim }_{x\to 1^{-}}f(x)=1+2=3$

• Right-hand limit:

${\lim }_{x\to 1^{+}}f(x)=2(1)+1=3$

• f(1) = 2(1) + 1 = 3

Since the limits and f(1) are equal, the function is continuous at x = 1.

No discontinuity here.

Example 2 – Removable Discontinuity

Given: $f(x)=\frac{x^2-4}{x-2}$

Determine if there is a discontinuity at x = 2, and classify it.

Solution:

Simplify the expression:

$f(x)=\frac{x^2-4}{x-2}$

For v $\ne$ 2: f(x) = x + 2.

But at x = 2:

$f(2)=\frac{4-4}{0}=\frac{0}{0}(\text{undefined})$

However,

${\lim }_{x\to 2}f(x)=2+2=4$

There is a Removable Discontinuity at x = 2.

Solved Example 3 – Infinite Discontinuity

Analyze the function: $f(x)=\frac{1}{x-3}$

Find the discontinuity.

Solution:

At x = 3:

• Left-hand limit:

${\lim }_{x\to 3^{-}}f(x)=-\infty$

• Right-hand limit:

${\lim }_{x\to 3^{+}}f(x)=+\infty$

The function has an Infinite Discontinuity at x = 3.

7.0Practice Questions on Discontinuity 

1. Check whether the function below is continuous at x = 2:

f(x)= $\{\begin{array}{ll}{x}^2-1,& x\le 2\\ 3x-2,& x>2\end{array}$

2. Identify the type of discontinuity (if any) in:

f(x)= $\{\begin{array}{ll}{x}^2-1,& x\le 2\\ 3x-2,& x>2\end{array}$

3. For which value of aa will the function be continuous everywhere?

f(x)= $\{\begin{array}{ll}2x+3,& x<1\\ a,& x=1\\ x^2+1,& x>1\end{array}$

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