How to Make a Function Not Continuous

Discontinuous Function

A function in algebra is said to be a discontinuous function if it is not a continuous function. Just like a continuous function has a continuous curve, a discontinuous function has a discontinuous curve. In other words, we can say that the graph of a discontinuous function cannot be made with a single stroke of the pen, i.e., once we put the pen down to draw the graph of a discontinuous function, we must pick it up at least once before the graph is complete. A discontinuous function has breaks/gaps on its graph and hence, in its range on at least one point.

Let us understand the concept of a discontinuous function, its definition and graph, and the types of discontinuous functions. We will also explore a few examples of a discontinuous function to understand better.

1.	What is a Discontinuous Function?
2.	Discontinuous Function Definition
3.	Discontinuous Function Graph
4.	Types of Discontinuous Function
5.	FAQs on Discontinuous Function

What is a Discontinuous Function?

A discontinuous function is a function in algebra that has a point where either the function is not defined at the point or the left-hand limit and right-hand limit of the function are equal but not equal to the value of the function at that point or the limit of the function does not exist at the given point. Discontinuous functions can have different types of discontinuities, namely removable, essential, and jump discontinuities. A discontinuous function has gaps along with its graph. In other words, we can say that if a function is not continuous, then it is called a discontinuous function. Discontinuous functions have holes or jumps in their graphs.

Discontinuous Function Definition

A function f is said to be a discontinuous function at a point x = a in the following cases:

The left-hand limit and right-hand limit of the function at x = a exist but are not equal.
The left-hand limit and right-hand limit of the function at x = a exist and are equal but are not equal to f(a).
f(a) is not defined.

The graph of a discontinuous function has at least one jump or a hole or a gap. Some of the examples of a discontinuous function are:

f(x) = 1/(x - 2)
f(x) = tan x
f(x) = x² - 1, for x < 1 and f(x) = x³ - 5 for 1 < x < 2

Discontinuous Function Graph

The graph of a discontinuous function cannot be made with a pen without lifting the pen. To draw a graph of a function that is discontinuous, once we put the pen down to draw the graph, we must pick it up at least once before the graph is complete and then continue to draw again. A discontinuous function has breaks or gaps on its curve. Hence, the range of a discontinuous function has at least one gap. We can identify a discontinuous function through its graph by identifying where the graph breaks and has a hole or a jump. In the next section, we will see the graph of a discontinuous function corresponding to different types of discontinuities.

Types of Discontinuous Function

Now that we know the definition of a discontinuous function, let us understand the different types of discontinuities of a function:

Removable Discontinuity: For a function f, if the limit lim _x _→a f(x) exists (i.e., lim _x _→a^- f(x) = lim _x _→a⁺ f(x)) but it is NOT equal to f(a). It is called 'removable discontinuity'.
Jump Discontinuity: For a function f, if the left-hand limit lim _x _→a^- f(x) and right-hand limit lim _x _→a⁺ f(x) exist but they are NOT equal. Hence, the limit if the function f does not exist. Then, x = a is called 'jump discontinuity' (or) 'non-removable discontinuity'.
Essential Discontinuity: The values of one or both of the limits lim _x _→a^- f(x) and lim _x _→a⁺ f(x) is ± ∞. It is called 'infinite discontinuity' or 'essential discontinuity'. One of the two left-hand and right-hand limits can also not exist in such discontinuity.

Important Notes on Discontinuous Function

A function that is not continuous is a discontinuous function.
There are three types of discontinuities of a function - removable, jump and essential.
A discontinuous function has breaks or gaps on its graph.

☛ Related Topics:

Limit Formula
Calculus
Types of Functions

Discontinuous Function Examples

Example 1: Identify if the function f(x) = (x - 2)/(x - 4) is a discontinuous function.

Solution: As we can see, the function f(x) = (x - 2)/(x - 4) is not defined at x = 4.

Hence it is discontinuous at x = 4.

Answer: f(x) = (x - 2)/(x - 4) is a discontinuous function.
Example 2: Find the point of discontinuity of the function f(x) = x² for x < 1, f(x) = 0 for x = 1, f(x) = 2 - (x - 1)² for x > 1.

Solution: The left-hand limit lim _x _→1^- f(x) = 1, and the right-hand limit lim _x _→1⁺ f(x) = 2

As we can the left and right-hand limits of the function f(x) are not equal, therefore f(x) is a discontinuous function and has a discontinuity at x = 1.

Answer: The point of discontinuity of f(x) = x² for x < 1, f(x) = 0 for x = 1, f(x) = 2 - (x - 1)² for x > 1 is x = 1.

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Discontinuous Function Questions

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FAQs on Discontinuous Function

What is a Discontinuous Function in Math?

A function f is said to be a discontinuous function at a point x = a in the following cases:

The left-hand limit and right-hand limit of the function at x = a exist but are not equal.
The left-hand limit and right-hand limit of the function at x = a exist and are equal but are not equal to f(a).
f(a) is not defined.

What Makes a Function Discontinuous?

A function is said to be a discontinuous function if any of the following cases is satisfied:

The left-hand and right-hand limits of the function at x = a exist but are not equal.
The left-hand limit and right-hand limit of the function at x = a exist and are equal but are not equal to f(a).
f(a) is not defined.

How Do You Identify a Discontinuous Function Using a Graph?

We can identify if a function is a discontinuous function using a graph if the graph has breaks, jumps, gaps, or holes.

What are the Types of Discontinuous Function?

There are three types of discontinuities of a discontinuous function, namely removable, jump, and infinite discontinuities.

Is a Discontinuous Function Differentiable?

If a function is discontinuous at a point, then automatically the function becomes not differentiable at that point. Hence, a discontinuous function is not differentiable at the point of discontinuity.

What is a Discontinuous Function Example?

An example of a discontinuous function is f(x) = 3/(2x - 4) as the function is not defined at x = 2.

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Source: https://www.cuemath.com/algebra/discontinuous-function/