How to know if a function is continuous at a point. Types of discontinuities

How do I know if a function is continuous at a point and what types of discontinuities exist and why do I need to know what kind of discontinuity it is?

In this lesson I explain how to know if a function is continuous at a point and I also tell you how many types of discontinuities exist and how they are identified.

In addition, we’ll be doing a few exercises on continuity of functions.

Function continuity: When is a function continuous?

To begin with, a function is continuous when it is defined in its entire domain, i.e. its domain is all R.

However, in certain functions, such as those defined in pieces or functions whose domain is not all R, where there are critical points where it is necessary to study their continuity.

A function is continuous at a point X0 if the limit exists when the function tends to that point and has a certain value and the value at that point is equal to the limit value:

how to tell if a function is continuous

In functions defined in pieces and in functions whose limit is an indeterminate number between zero, in order for the limit of the function to exist at a point, we must calculate the lateral limits and these must have the same value, that is, the limit on the left and on the right of that point must coincide:

when a function is continuous

If all the above is not met, the function will not be continuous, that is, if the limit exists but does not coincide with the value of the function or the limit does not exist at that point or the function does not exist at that point, the function will not be continuous and therefore, there will be one of the types of discontinuities that I will explain in the following section.

Types of discontinuities

Depending on the condition that is not met for a function to be continuous at one point, we will have different types of discontinuities. They are as follows:

Avoidable discontinuity

A function is not continuous at one point and will have an avoidable discontinuity when the limit of the function at that point exists, but the value of the function at that point is different from the limit value:

funcion continua

The graphical representation of a function with an avoidable discontinuity, when the function at that point has a value other than the limit value is for example:

how to tell if a function is continuous or discontinuous

It is also an avoidable discontinuity when there is the limit of the function at one point, but the function at that point does not exist:

how to tell if a function is continuous

The representation in Cartesian axes of a function with an avoidable discontinuity when the value of the function at a point does not exist, but the limit would be, for example:

where is the function continuous and why?

As an anecdote, avoidable discontinuity is so called, because it could be avoided if the value of the function at the critical point existed and was equal to the function of the limit.

Unavoidable discontinuity of finite jump

A function is not continuous at one point and will have an unavoidable discontinuity of finite hop when the limit of the function when x tends to that point does not exist, i.e. the lateral limits each result in a different finite value and therefore do not coincide:

when a function is continuous

The representation of a function with an unavoidable finite-hop discontinuity can be for example:

how to determine if a function is continuous

Infinite hopping discontinuity not avoidable

A function will have an unavoidable discontinuity of infinite jump and therefore will not be continuous at that point, when the limit of the function when x tends to that point does not exist, or what is the same, that the lateral limits result in an infinite value and therefore do not coincide:

when a function is continuous

The graphical representation of a function with an unavoidable discontinuity of infinite jump can be for example this one:

know if a function is continuous

It may be the case that the limit exists, since both lateral limits have the same result that is “more infinite” or “less infinite”, but in this case, the function at that point would not exist, as there is an asymptote and therefore the function would not be continuous at that point.

Resolved function continuity exercises

We are now going to solve some exercises on continuity of functions step by step to apply what we have learned so far.

Exercise on continuity of functions 1

It studies the continuity of the next function in x=3 and if it is not continuous, indicate what type of discontinuity it is:

a function is continuous when

To check if this function is continuous in 3 we must calculate the limit of the function when x tends to 3, if it exists and then calculate the value of the function in x=3.

We begin by calculating the limit of the function when x tends to 3:

how to tell if a function is continuous at a point

We replace the x with a 3 and it gives us an indeterminacy of zero to zero:

how to know when a function is continuous

This type of indeterminacy is resolved by breaking down the numerator and denominator into factors:

how to tell if a function is discontinuous

We eliminate the factor (x-3) as it is repeated up and down. We’ve got some left:

continuous and discontinuous function

We replace the x with 3 again and reach the limit result:

how to know if a function is continuous

The limit of the function when x tends to 3 is equal to 3.

Let’s now calculate the value of the function in x=3. To do this in the function, we replace the x with 3 and we are left with it:

determine if a function is continuous

In this case, the result is also zero to zero, but since it is not a limit, this result does not exist. Therefore there is no function at x=3.

The limit of the function when x tends to 3 is 3 and the function in x=3 does not exist, therefore the function is not continuous in x=3:

when a function is not continuous

This is an avoidable discontinuity.

If we wanted to redefine the function so that it would be continuous we would do it this way:

how to know that a function is continuous

For any value of x other than 3, the function takes the value of the first section of the function, but when x=3, as the function of the first section is not defined for that value, we give it the value 3.

This way the function is continuous.

Let’s figure out another exercise.

Exercise on continuity of functions 2

Study the continuity of the next function:

continuous and discontinuous functions

In this case they do not tell us the points where we have to study the function, but as it is a function defined in pieces, we must study its continuity in its critical points, that is to say, in the points where the function changes sections, which in this case are -1 and 2.

We start with -1.

First we will calculate the limit when x tends to -1. As it is a function defined in pieces, to calculate the limit of the function when x tends to -1, we must calculate its lateral limits.

Let’s calculate the limit of the function when x tends to -1 on the left:

continuous function

The left of -1 are the values of x smaller than -1. The function exists for values smaller than -1 in the first leg, therefore, is the one we have to use to calculate the limit:

demonstrate that a function is continuous

We replace the x with -1 and arrive at the result of the limit which is equal to -3:

which is a continuous function

We keep calculating the limit of the function when x tends to -1 on the right:

when it is a continuous function

The right of -1 are the values of x greater than -1. The function exists for values greater than -1 in the second leg, therefore, is the one we have to use to calculate the limit:

as if a function is continuous

Directly, as there is no x to substitute, the limit value when x tends to -1 to the right is -3.

The two lateral limits coincide, therefore, the limit of the function when x tends to -1 exists and is equal to -3:

how to know if the function is continuous

The first condition for the function to be continuous at -1 is fulfilled.

Let’s now calculate the value of the function in x=-1. The section of the function that we have to use to calculate the value of the function in x=-1 is the second section, since it is where the -1 has the equal sign (the second section exists for values greater than or equal to -1):

determine if the function is continuous

The function in x=-1 is equal to -3, which coincides with the limit of the function when x tends to -1, therefore, the function is continuous in x=-1.

We continue to study the continuity of the function at x=2.

The first step is to calculate the limit when x tends to 2. As it is a function defined in pieces, its lateral limits must be calculated.

Let’s calculate the limit of the function when x tends to 2 from the left:

how to tell if it is continuous or discontinuous

The function exists for values that are less than 2 in the second section, so it is the section that we use to calculate this lateral limit, whose result gives us -3:

when a function is continuous at a point

We calculate the limit of the function when x tends to 2 from the right:

how to show that a function is continuous

To calculate this limit, we use the section of the function that is defined for values of x greater than 2, which is the third section:

how to see if a function is continuous

We replace the x with 2 and get the limit value:

single point continuous function

The lateral limits do not coincide, so the limit of the function when x tends to 2 does not exist:

which means that a function is continuous

Therefore, the function is not continuous at x=2 and is an unavoidable discontinuity of finite hop.

It is no longer necessary to calculate the value of the function for x=2.