Discontinuous Function

The diagram of a discontinuous function cannot be made with a single stroke of the pen, which means 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.

Related Topics

• How to Remove Discontinuities
• How to Find Continuity at a Point

A step-by-step guide to a discontinuous function

A discontinuous function is a function in algebra that has a point at which either the function is not defined at that point or the left and right-hand limits 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.

A discontinuous function has gaps along with its graph. In other words, if it is not a continuous function, it is called a discontinuous function. Discontinuous functions have holes or jump 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.

Discontinuous function graph

The diagram of a discontinuous function cannot be made with a pen without lifting the pen. To draw a graph of a function that is discontinuous, when we put the pen down to draw the graph, we must pick it up at least once before completing the graph and then continue drawing again.

A discontinuous function has breaks or gaps on its curve. Therefore, the graph of a discontinuous function has at least one break, hole, or jump. We can identify a discontinuous function through its diagram by identifying where the diagram is broken and has a hole or a jump.

Types of a discontinuous function

The different types of discontinuities of a function are:

• Removable discontinuity:

For a function f, if the limit \(lim _{x\to a}\:f\left(x\right)\) exists (i.e., \(lim_{x\to a^-}\:f\left(x\right)=lim_ {x\to a^+}\:f\left(x\right)\)) but it is not equal to \(f(a)\). It is called removable discontinuity.

• Jump discontinuity:

For a function f, if the left-hand limit \(lim _{x\to a^-}\:f\left(x\right)\) and right-hand limit \(lim _{x\to a^+}\:f\left(x\right)\) exist but they are not equal. Hence, the limit of 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 \(lim _{x\to a^-}\:f\left(x\right)\) and \(lim _{x\to a^+}\:f\left(x\right)\) is \(\pm \:∞\). 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.

Discontinuous Function , Example 1:

Identify whether the function \(f(x) = \frac{(x – 3)}{(x – 6)}\) is a discontinuous function.

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Solution:

 The function \(f(x) = \frac{(x – 3)}{(x – 6)}\) is not defined at \(x = 6\). So, it is discontinuous at \(x = 6\).

Exercises for Discontinuous Function 

Find the point of discontinuity of each function.

1. \(\color{blue}{f(x)=\frac{x^2-4x-21}{x+3}}\)
2. \(\color{blue}{f(x)=\frac{x^2-x-2}{x^2-5x-6}}\)
3. \(\color{blue}{f(x)=\frac{2x+3x^2+x^3}{x^2-x}}\)
4. \(\color{blue}{f\left(x\right)=\frac{\sqrt{x^2+5}}{x+5}}\)

1. \(\color{blue}{x=-3}\)
2. \(\color{blue}{x=-1, 6}\)
3. \(\color{blue}{x=0, 1}\)
4. \(\color{blue}{x=-5}\)

The three-step continuity test, with examples

A function f is continuous at x = a only if all three of these are true:

1. f(a) is defined.
2. The limit of f(x) as x approaches a exists, which means the left and right limits are equal.
3. That limit equals f(a).

Example 1, a hole: f(x) = (x² − 9)/(x − 3). At x = 3 the denominator is 0, so f(3) is undefined and step 1 fails. Factor the top: (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3. The limit is 3 + 3 = 6, so the graph has a hole at (3, 6). This is a removable discontinuity.

Example 2, a jump: f(x) = x + 1 when x < 2, and f(x) = 5 when x ≥ 2. From the left, f(x) approaches 2 + 1 = 3. From the right, f(x) is 5. The one-sided limits differ, so step 2 fails and the graph jumps at x = 2.

Types of discontinuity compared

Jump and infinite discontinuities are both non-removable: no single point can fill the gap.

Frequently Asked Questions

What is a discontinuous function?

A discontinuous function is a function whose graph has at least one break, such as a hole, a jump or a vertical asymptote, so it cannot be drawn without lifting your pencil. It is still a real function, because each input in its domain has exactly one output. The break simply means it is not continuous at that point.

How do you tell if a function is discontinuous?

Use the three-part continuity test at x = a. Ask whether f(a) is defined, whether the limit of f(x) as x approaches a exists, and whether that limit equals f(a). If any answer is no, the function is discontinuous at a. For instance, f(x) = 1/(x − 2) fails the first test at x = 2 because f(2) is undefined.

What is an example of a discontinuous function?

A simple example is f(x) = 1/x, which has a vertical asymptote at x = 0 and is undefined there. Another is the floor function, which rounds down to the nearest integer and jumps up by 1 at every whole number. A piecewise rule such as f(x) = 1 for x < 0 and f(x) = 3 for x ≥ 0 also jumps at x = 0.

What is a function in which the points on the graph are not connected called?

A function whose graph is made of separate, unconnected points is called a discrete function. An example is the cost of concert tickets at $15 each, since you can only buy 0, 1, 2 or 3 tickets, giving the points (1, 15), (2, 30), (3, 45). A discontinuous function is different: its graph has connected pieces with a hole, jump or asymptote between them.

How do you find where a function is discontinuous?

Look for x-values that make a denominator zero, boundaries in a piecewise rule, and places where a step function jumps. For a rational function, factor first. In (x² − 9)/(x − 3), the top factors as (x − 3)(x + 3), so the factor cancels and leaves a hole at x = 3. A factor that stays in the denominator gives a vertical asymptote instead.

What are the types of discontinuity?

There are three main types. A removable discontinuity is a single hole, where the limit exists but the function is undefined there or has a different value. A jump discontinuity happens when the left-hand and right-hand limits are different numbers. An infinite discontinuity occurs at a vertical asymptote, where the function grows without bound, as 1/(x + 4) does near x = −4.