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111 LearnersLast updated on October 22, 2025
Discontinuous Function
As the name suggests, a discontinuous function is not continuous and does not have a smooth graph. A discontinuous function has gaps or breaks on its graph and has at least one point in its range. In this article, we will learn more about the discontinuous function.
What is a Discontinuous Function?
A discontinuous function is not continuous, and when represented on a graph, we might see disruptions in the graph line. The disruptions may appear in the form of breaks, jumps, or holes. Discontinuity occurs when the function suddenly jumps, is not defined at a certain value of x, or has a missing point. Discontinuous functions exhibit various types of discontinuities, including removable, essential, and jump discontinuities. Now, what do we mean by these? A removable discontinuity happens when the function is either undefined or not equal to the limit at a certain x-value, leaving a hole in the graph. A jump discontinuity is when the function suddenly jumps from one value to another. Finally, essential or infinite discontinuity occurs when the function is not defined.
Difference Between Continuous and Discontinuous Functions
The difference between continuous and discontinuous functions are given below:
|
Feature |
Continuous Function |
Discontinuous Function |
|
Definition |
In a continuous function, we can draw the whole graph without lifting the pencil. |
We have to lift the pencil at some point while drawing the graph. |
|
Graph |
It is a smooth line or curve with no holes, jumps, or breaks. |
The line has breaks, jumps, or holes in it. |
|
Limit |
The value of the function at a point is the same as the value we get when we get close to that point. |
The value might be undefined or may not approach the same number from both sides as we get closer to that point. |
| Derivative |
We can find the slope at every point in the function’s domain. |
The slope might not exist at some point. |
|
Integration |
In a continuous function, we can find the area under the curve on any part of the graph. |
Sometimes we can’t find the area under the curve at some point. |
|
Examples |
Normal smooth functions like: |
Functions that jump or break like: |
How to Represent Discontinuous Function in Graph?
A discontinuous function is a function whose graph cannot be drawn in one smooth line. This happens because the graph has holes, breaks, or jumps in it, leading to missing values from the range. To spot the discontinuous function, just look at the graph and see if there is a hole, a sudden jump, or a break anywhere.
How to Identify Discontinuities?
The gaps or breaks in a discontinuous function are called discontinuities. There are three common discontinuities, and they are:
- Removable discontinuity
- Jump discontinuity
- Essential discontinuity
Removable discontinuity: A hole in the graph at a certain x-value where the function is not defined is known as a removable discontinuity. The graph will be smooth, but only one point will be missing. We can fix it by filling the hole. Imagine if we drop a tiny dot from a line; that is a removable discontinuity.
Jump discontinuity: Here, the graph suddenly jumps from one value to another, leaving a gap between the left and right sides. The function has different values as we approach the point from each side, and the graph won’t connect at that point.
Essential discontinuity: An essential discontinuity occurs when the function keeps changing its direction or approaches positive or negative infinity near a point. There is no way of fixing essential discontinuity. For example, think of a roller coaster that zooms up and down at one point.
Common Mistakes and How To Avoid Them in Discontinuous Function
Working on problems involving discontinuous function can be tricky, and we may even end up making mistakes. However, these mistakes can be avoided if we practice regularly and keep ourselves alert. Here are a few common mistakes which can be easily avoided.
Mistake 1
Only checking the limit from one side
Sometimes, we forget to check the limit from both sides and check from just one direction, which can lead to the wrong conclusion about whether a function is continuous. Always check the left-hand limit and right-hand limit at the point of intersection. If both sides are not equal, then the function has a jump discontinuity.
Mistake 2
Concluding all the graphs with a break as a jump discontinuity
It’s common to assume that any break in a graph means a jump discontinuity, but that’s not always true. Not all the breaks are jumps. Some breaks are just holes, which can be removable. Learn to analyze limits and function values carefully.
Mistake 3
Checking the graph visually without using limits
Students may assume a function is continuous just because the graph looks connected. Use the limits to confirm. Even if the graph appears smooth, the function could still have a discontinuity that isn’t visible at first glance.
Mistake 4
Believing all piecewise functions are discontinuous
Not all piecewise functions are discontinuous. They can still be continuous if different pieces of the function meet somewhere and the right-hand limit, left-hand limit, and function value all match. For example, the function given below is discontinuous at x = 1 because the right and left limits are not equal.
\(f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 1 & \text{if } x = 1 \\ x + 0 & \text{if } x > 1 \end{cases} \)
But we can define it as:
\(f(x) =
\begin{cases}
x^2 & \text{if } x \leq 1 \\
x^2 & \text{if } x > 1
\end{cases}
\)
When defined like this, it then becomes a continuous piecewise function as the pieces connect without any disruptions.
Mistake 5
Ignoring the function’s actual value at a point
Even if a function has matching limits, it might still be discontinuous if the value at that point is missing or different.
Real Life Applications of Discontinuous Function
In real life, discontinuous functions are used to model sudden or abrupt changes. Here are some real-life applications of a discontinuous function:
- Billing or Pricing: Many companies use step or piecewise functions to decide costs. For example, mobile companies use this to charge their customers methodically. When a customer exceeds 100 minutes of free talk-time, they will automatically be charged for any additional minutes. This jump in the price can be modeled using a step or piecewise function.
- Elevator: Elevators’ movement between floors can be modeled using a step function. An elevator doesn’t stop between floors, so the graph of its position would show sudden jumps, representing a jump discontinuity.
- Traffic Lights: Since traffic lights change instantly without any gradual transition, this shift can be modeled using a step function.
Solved Examples on Discontinuous Function
Problem 1
Is the function x2 - 4x - 2 continuous at x = 2?
No, the function is discontinuous at x = 2.
Explanation
The function can be written as,
(x - 2)(x + 2)/x - 2
After canceling, the function becomes: x - 2. Therefore, f(x) = x + 2, but only when x ≠ 2. At x = 2, the original function is undefined, so there is a hole in the graph, and it is a removable discontinuity.
Problem 2
Is f(x) = 1x continuous at x = 0?
No, it is not continuous at x = 0.
Explanation
When x gets close to 0 from the left side, the values go to negative infinity.
When x gets close to 0 from the right side, the values go to positive infinity.
The function is also not defined at x = 0.
So, there is a wild change at 0, this is called an essential discontinuity.
Problem 3
If a graph has a hole at x = 3, is the function continuous here?
No, it is discontinuous at x = 3.
Explanation
A hole means there is no value at that point.
If the graph is smooth around it, the missing value makes it not continuous.
That’s a removable discontinuity because we can fix it by putting a value in the hole.
Problem 4
Is f(x) = 1x - 5 continuous at x = 5?
No, it is not continuous at x = 5.
Explanation
At x = 5, the denominator becomes 0, so the function is not defined.
As x gets close to 5, the function shoots up or down to infinity.
That’s an essential discontinuity.
Problem 5
Is the function f(x) = x2 - 1x - 1 continuous at x = 1?
No, it is not continuous at x = 1.
Explanation
Factor the numerator: x2 - 1 = (x - 1)(x + 1)
Cancel x - 1: f(x) = x + 1, but only when x ≠ 1.
At x = 1, we would divide by 0, which is not allowed. So the function is undefined here.
The graph has a hole at x = 1. That is a removable discontinuity.
FAQs on Discontinuous Function
1.What is a discontinuous function?
2.Can a function be discontinuous at more than one point?
3.What is a removable discontinuity?
4.What is a jump discontinuity?
5.Is a piecewise function always discontinuous?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
