Discontinuous Functions: Understanding the Types and Real-Life Applications

Functions are an essential concept in mathematics that play a crucial role in describing the relationship between input and output values. However, not all functions behave in the same way. Continuous functions are functions where the output changes smoothly and without abrupt jumps as the input varies within a given interval. In other words, the graph of such a function can be drawn without lifting the pen from the paper. Whereas some functions exhibit discontinuities, or breaks which can pose challenges in analysing their behaviour and applications. In this maths article, we will explore what discontinuous functions are, their types, and real-world applications.

What is a Discontinuous Functions?

A function is a discontinuous function if one of the following is true about the function:

• There’s a point where the function is not defined at the point 
• There’s a point where the left-hand limit and right-hand limit of the function are equal but not equal to the value of the function.
• There’s a point where the limit of the function does not exist.

A function is discontinuous at a point if it fails to be continuous at that point. In other words, a discontinuous function has one or more points where its graph has a break, hole, or jump. A discontinuity occurs when a function's value changes abruptly or is undefined at a particular point. These discontinuities can arise due to various reasons, including limits, jumps, and asymptotic behaviour.

Discontinuous Function Definition

A function is called discontinuous at a point x = a if something goes wrong in how the graph behaves at that point. This can happen in a few ways:

1. The left and right limits exist but are not equal
   This means the graph approaches two different values from each side of x = a. So, there’s a jump in the graph.

2. The left and right limits are equal, but not equal to the actual value of the function at that point (f(a))
   This creates a hole in the graph—because the function "should" be at a certain height, but it's not.

3. The function isn’t defined at that point (f(a) doesn’t exist)
   That means there's a gap or the graph is broken at that spot.

So, if any of these happen, the function is not smooth at that point—it’s called discontinuous.

Discontinuous Function Examples

Let us look at some examples of discontinuous functions:

Absolute value function: f(x) = |x|

The absolute value function has a jump discontinuity at x = 0 because the function changes direction abruptly from negative values to positive values at that point.

Step function: f(x) = ⌊x⌋

The step function has a jump discontinuity at every integer value of x because the function jumps from one integer to the next, without taking any values in between.

Dirichlet function: f(x) = { 1 if x is rational; 0 if x is irrational }

The Dirichlet function is discontinuous at every point in its domain because it takes the value of 1 for rational values of x and 0 for irrational values of x.

Rational function: f(x) = 1 / (x - 1)

The rational function has an essential discontinuity at x = 1 because the limit of the function as x approaches 1 does not exist.

Piecewise-defined function: f(x) = { x + 1 if x < 0; x^2 if x ≥ 0 }

The piecewise-defined function has a jump discontinuity at x = 0 because the function changes from a linear function to a quadratic function at that point.

These are just a few examples of discontinuous functions. Discontinuous functions can take many different forms and have different types of discontinuities, such as jump discontinuities, removable discontinuities, and essential discontinuities.

Graphs of Discontinuous Functions

The graph of a discontinuous function shows breaks or interruptions in the curve. At points of discontinuity, the function may have holes, jumps, or asymptotic behaviour. The graph may appear fragmented or irregular, with different segments having different values. However, these discontinuities can provide important information about the behaviour and properties of the function, such as limits, derivatives, and integrals. Understanding the graph of a discontinuous function is essential in many mathematical applications, such as signal processing, optimization, and control theory.

Types of Discontinuity

Discontinuities come in three main types: removable, jump, and essential discontinuities. Removable discontinuities occur when there is a hole in the graph that can be filled in by defining the function at that point. Jump discontinuities arise when the function jumps from one value to another at a specific point. Essential discontinuities occur when a function approaches a value that it never achieves, and the limit does not exist.

Let's consider some examples to better understand the different types of discontinuity:

Removable Discontinuity

A removable discontinuity occurs when a function is undefined at a particular point, but the hole can be "filled" by assigning a value to the function at that point. In other words, the function can be made continuous at that point by redefining it to have a value equal to its limit as x approaches the point of discontinuity. A removable discontinuity is called "removable" because the hole in the graph can be removed by redefining the function at the point of discontinuity.The function f(x) =(x2−4)/(x−2) has a removable discontinuity at x = 2. The denominator becomes zero at x = 2, causing an undefined value. However, we can fill in the hole by defining f(2) = 4. The graph of the function has a hole at x = 2.

Jump Discontinuity

A jump discontinuity is a type of discontinuity that occurs when the limit of a function from the left-hand side and the right-hand side at a particular point exists, but are not equal. In other words, the function has a sudden "jump" in value at that point, resulting in a gap or jump in the graph of the function. The size of the jump is equal to the difference between the limits from the left and right-hand side. A common example of a function with a jump discontinuity is the step function, which is defined as follows:

f(x) = 0 for x < 0

f(x) = 1 for x >= 0

At x = 0, the limit from the left is 0, while the limit from the right is 1. Therefore, the function has a jump discontinuity at x = 0, with a jump of size 1. 

Other examples of functions with jump discontinuities include the floor function and the ceiling function.

Essential Discontinuity

An essential discontinuity is a type of discontinuity that occurs when the limit of a function at a particular point does not exist. In other words, the function behaves in a completely unpredictable way near that point. Essential discontinuities are also known as infinite discontinuities because the function either goes to positive or negative infinity or oscillates wildly without converging to any limit.

A common example of a function with an essential discontinuity is the function f(x) = 1/x, which has a vertical asymptote at x = 0. As x approaches 0 from the right, the function becomes increasingly large and positive, while as x approaches 0 from the left, the function becomes increasingly large and negative. Therefore, the limit of the function at x = 0 does not exist, and the function has an essential discontinuity at that point.

The function f(x) = sin(1/x) has an essential discontinuity at x = 0. As x approaches 0, sin(1/x) oscillates wildly between -1 and 1, and the limit does not exist. The graph of the function becomes increasingly oscillatory as x approaches 0.

Identifying Discontinuities

To identify discontinuities, we can examine the function's graph or algebraically by computing limits. A removable discontinuity appears as a hole in the graph that can be filled in by defining the function at that point. A jump discontinuity appears as a sudden change in the graph from one value to another at a specific point. An essential discontinuity occurs when the limit does not exist or approaches a value that the function never achieves.

To identify a removable discontinuity, you need to examine the graph of the function and look for a hole or gap where the function is undefined at a particular point. This can be done by finding the limit of the function as it approaches that point. If the limit exists and is finite, but is different from the value of the function at that point, then the function has a removable discontinuity at that point. In other words, the function has a hole in its graph that can be filled by redefining the function at the point of discontinuity.

One way to determine whether a function has a removable discontinuity is to plot its graph and look for a hole or gap in the curve. Another method is to factor the function and see if there is a common factor that can be cancelled out to eliminate the point of discontinuity. For example, in the function f(x) = (x2−4)/(x−2), the point of discontinuity occurs at x = 2 because the denominator becomes zero at that point. However, if we factor the numerator, we get f(x) = (x - 2)(x + 2)/(x - 2), which simplifies to f(x) = x + 2, except at x = 2, where the original function had a hole. Therefore, the function has a removable discontinuity at x = 2, which can be filled by redefining f(2) to be 4, the value of the limit as x approaches 2.

To identify a jump discontinuity, you need to examine the graph of the function and look for a sudden "jump" or gap in the curve at a particular point. This can be done by finding the limit of the function as it approaches that point from both the left and the right sides. If the left and right-hand limits exist but are not equal, then the function has a jump discontinuity at that point. In other words, the function changes its value abruptly at that point, resulting in a gap or jump in the graph of the function.

One way to determine whether a function has a jump discontinuity is to plot its graph and look for a vertical gap or jump in the curve. Another method is to examine the limits of the function from both the left and right sides at the point of discontinuity. For example, in the function f(x) = |x|, point of discontinuity occurs at x = 0 because the limit from the left is -1, while the limit from the right is 1. Therefore, the function has a jump discontinuity at x = 0, with a jump of size 2.

It is important to note that not all functions with vertical gaps or jumps in their graph have jump discontinuities. For example, the function f(x) = 1/x has a vertical asymptote at x = 0, but it does not have a jump discontinuity because the left and right-hand limits do not exist. Therefore, to identify a jump discontinuity, it is essential to check whether the left and right-hand limits exist and are not equal at the point of discontinuity.

To identify an essential discontinuity, you need to examine the limit of the function as it approaches the point of discontinuity from both the left and the right sides. If the limits do not exist or if they approach different values, then the function has an essential discontinuity at that point. In other words, the function behaves in a completely unpredictable way near that point, either by going to positive or negative infinity or oscillating wildly without converging to any limit.

One way to determine whether a function has an essential discontinuity is to plot its graph and look for a vertical asymptote or a wildly oscillating behaviour near the point of discontinuity. Another method is to examine the limits of the function from both the left and right sides at the point of discontinuity. For example, in the function f(x) = 1/x, the point of discontinuity occurs at x = 0 because the limit from the left is negative infinity, while the limit from the right is positive infinity. Therefore, the function has an essential discontinuity at x = 0.

Let's look at a tabular comparison between continuous and discontinuous functions:

I hope this table helps you understand the main differences between continuous and discontinuous functions.

Discontinuous Function in Fourier Series

In simple terms, the Fourier series is a way to represent a function as an infinite sum of trigonometric functions, such as sine and cosine. However, for a discontinuous function, the Fourier series may not always give an accurate approximation of the function.

For example, consider the function f(x) = |x| on the interval [-π, π]. This function has a jump discontinuity at x = 0, where the left and right limits are not equal. When we try to represent this function using a Fourier series, the series will converge to a value between the left and right limits of the function at x = 0, but the series will oscillate near the point of discontinuity, creating a phenomenon known as the Gibbs phenomenon.

Another example is the function f(x) = 1/x on the interval [1, 2]. This function has an essential discontinuity at x = 1, where the function is not defined. In this case, the Fourier series will not converge uniformly to the function over the interval, and the error in the approximation may be large.

However, some discontinuous functions can be approximated well by Fourier series, such as the sawtooth wave, which has a continuous derivative but a jump discontinuity. In general, the behaviour of the Fourier series of a discontinuous function depends on the type and severity of the discontinuity and may require specialised techniques to analyse.

Applications of Discontinuous Functions

Discontinuous functions play a crucial role in signal processing, where they are used to analyse and manipulate digital signals. One such example is the staircase function, which is a series of discrete steps. This function is used to represent sampled signals in digital signal processing applications.

Let's consider an example. Suppose we want to convert an analog signal (e.g., a sound wave) to a digital signal (e.g., an MP3 file). We first sample the analog signal at regular intervals, producing a sequence of discrete values. We can represent this sequence using a staircase function, where the steps correspond to the sample values. The resulting digital signal can then be processed, stored, and transmitted using various signal processing techniques.

Another application is in control theory, where step functions are used to simulate the behaviour

Discontinuous function solved example

Key Points About Discontinuous Functions

• A function is called discontinuous if it is not smooth or not connected at some point.
  In short: if it’s not continuous, it’s discontinuous.
• There are three main types of discontinuities:
  1. Removable discontinuity – a small hole in the graph that can be “fixed” by redefining the function.
  2. Jump discontinuity – when the graph jumps from one value to another.
  3. Essential (or infinite) discontinuity – when the graph goes off to infinity or has a very sharp break.
• A discontinuous function will show gaps, holes, or sudden jumps in its graph.

Discontinuous Functions Examples

Example 1: Determine whether the function f(x) = is continuous at x = 2.

Solution:

To determine if the function f(x) = is continuous at x = 2, we need to check if the left and right limits at x = 2 are equal to the value of the function at x = 2.

Left limit: lim x → 2- = 4

Right limit: lim x → 2+ = 4

Function value at x = 2: f(2) = 4

Since the left and right limits are both equal to the function value at x = 2, the function f(x) = is continuous at x = 2.

Example 2: Determine the point of discontinuity for the function
f(x) = (x² − 1) / (x − 1)

Solution:
First, factor the numerator:
x² − 1 = (x − 1)(x + 1)

So,
f(x) = [(x − 1)(x + 1)] / (x − 1)

For all x ≠ 1, the expression simplifies to:
f(x) = x + 1

However, at x = 1, the denominator becomes zero, so the function is undefined at that point.

Since the limit of f(x) as x approaches 1 exists (it is equal to 2), but the function is not defined at x = 1, this is called a removable discontinuity.

Therefore, the function has a removable discontinuity at x = 1.

Example 3: Determine whether the function f(x) = sin(x) / x  is continuous at x = 0

Solution:
The function f(x) = sin(x) / x is undefined at x = 0 because of division by zero.

However, the limit exists:

lim (x → 0) [sin(x) / x] = 1

If we define f(0) = 1, then the function becomes:

f(x) =
  sin(x) / x, if x ≠ 0
  1,     if x = 0

Now, the function is continuous at x = 0 because:

• The left-hand and right-hand limits are both equal to 1
• The function value at x = 0 is also 1

Therefore, the function has a removable discontinuity at x = 0, which can be removed by defining f(0) = 1.

Note: Even though the function is not continuous at x = 0, it has a removable discontinuity. This is because the function can be redefined at x = 0 to make it continuous by setting f(0) = 1, which is the limit of the function as x approaches 0.

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