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Continuity

One way to look at continuity is to start with the technical definition, and then do a bunch of examples to see what is or isn't continuous using the definition. Instead, we will begin with an intuitive understanding of continuous functions and build what we would like to see in the definition.

Intuitively Continuous

You have probably heard someone say that "a function is continuous if you can draw it without picking up your pencil". So let's look at what needs to be true for that to happen, starting at a single point p.

If you try and draw this function without picking up your pencil you can't because the function has a hole at p. In fact, it isn't even defined at p! So you are definitely going to need to assume that the function you are thinking about is defined at the point p, and this is done by saying "assume exists".

This function is not defined at the point p | StudySmarter Originals

But is the function having a value at p enough to make sure you can draw it without picking up your pencil? Let's look at another case.

This function is defined at p, but the limit from the left and the right aren't the same. In other words

so

does not exist. So, the limit as x approaches p must exist as well!

The limit from the left and right are not the same | StudySmarter Originals

So even having the function defined isn't enough. You also need the limit from the left and right to have the same value. But is that enough to make sure you can draw it without picking up your pencil? Let's see!

The function below is defined at p. The limit exists as x approaches p. But it isn't equal to the function value! You can write these two values not being the same as

The function value isn't the same as the limit of the function | StudySmarter Originals

Now let's put all of that together into a definition.

Continuity Definition

The function is continuous at the point p if and only if all the following three things are true:

1. exists

2. exists (the limit from the left and right are equal)

3.

If a function fails any of those three conditions, then is said to be discontinuous at p, or simply not continuous at p.

The expression "if and only if" is a biconditional logic statement, meaning if A is true, then B is true, and if B is true, then A is true.

Easy Steps to check if a Function is Continuous

You can use the definition to make a step-by-step process to check and see if a function is continuous at p.

Step 1: Make sure the function is defined at p. If it isn't, stop because the function definitely isn't continuous at p.

Step 2: Make sure that exists. If it doesn't, you can stop because the function definitely isn't continuous at p.

Step 3: Make sure that the limit and the function value are equal. If they aren't, then the function definitely isn't continuous at p.

Note that sometimes if a function is discontinuous at a point, people will say that it has a discontinuity at that point. These two phrases mean the same thing.

Continuous Function Examples

Let's practice determining if a function is continuous at a certain point!

Decide whether or not the function

is continuous at .

If you try and evaluate the function at 2, you get division by zero. So, in fact, this function is not defined at , so it can't be continuous at that point either. You can graph the function to see that there is a vertical asymptote at , which is why it isn't defined there.

This function is not continuous at x=2 | StudySmarter Originals

So the difficulty with the previous example was that the function wasn't defined when . Suppose, instead, your function is defined by

which is definitely defined at Is this function continuous at ?

In this case

but

,

so the limit doesn't exist at . Therefore, even though the function is defined when , it is not continuous there.

Let's rig the previous example some more. If the problem is that the limit from the left and right aren't the same, you can change the function a bit to see what happens. Take

which is still defined when . Now is the function continuous at

Now when you look at the limit as approaches 2, you have

But , which is certainly not infinity! So the function is still not continuous at .

Decide whether the function

is continuous when .

The trick here is to read the question carefully. We don't necessarily look at the point where the function changes definition, we look at what the question is asking!

In this case, the function changes definition at x=2, but we are asked if it is continuous at x=3. So all you need to consider is if the function is continuous at . But this is just a line, so you know

,

and the function is continuous at .

Let's work it out step by step:

Step 1: Make sure the function is defined at . . Therefore, it is defined.

Step 2: Make sure the limit at exists. That is, check if the limit from the left and right of are equal.

and .

The limits from the left and right of are indeed equal.

Step 3: Lastly, check if the limit is equal to the function value at . That is:

This last condition is satisfied. Therefore, the function is continuous at .

In the previous example, the point p wasn't where the function had a switch in which formula was used. What if, instead, the point you cared about was ?

Decide whether the function

is continuous at the point .

Let's follow the same steps as the previous example.

Step 1: Check if the function is defined at .

.

Therefore, the function is defined at .

Step 2: Now you check if the limit exists. The limit from the left gives

and the limit from the right is

which means that

Step 3: Lastly, check if the function value from step 1 and limit from step 2 are equal. They both equal 5!

So, you've checked:

1. That the function is defined at the point,
2. The limit of the function exists at that point, and
3. The function value at that point has the same value as the limit.

Therefore, the function is continuous at .

What if we changed the function in the previous example slightly?

Decide whether the function

is continuous at the point .

Step 1:

Just like in the previous example,

Step 2: Check left and right-hand limits:

The limit from the left:

But now the limit from the right is

so

does not exist. Therefore the function is not continuous at the point .

Here, since the criterion in step 2 is not met, we don't have to continue on to step 3!

Continuity in Calculus

Why should you care whether or not a function is continuous? Suppose that you are modeling a population with x measured in years, and you find that the formula for it is given by

which from the work you did above is not continuous at . Taking a look at the graph of this function,

A piecewise function which is not continuous at p=2 | StudySmarter Originals

Knowing that the function is not continuous at lets you know that something drastic happened to the population you're studying. In this case, it is a sudden die-off, which is a problem you would want to investigate.

Types of Continuity

Here you have looked explicitly at continuity at a point. But what about continuity over an interval, or even the whole real line? For information on intervals, see Continuity over an Interval, and for more theorems on continuity, see Theorems of Continuity.

Continuity - Key takeaways

• Intuitively, a function is continuous if you can draw it without picking up your pencil.
• The function is continuous at the point if and only if the function is defined at , the limit of the function exists at , and the function value and the limit at both have the same value.
• A function that is not continuous at the point is said to be discontinuous.
• A function fails to be continuous at the point if
• the function isn't defined there, or
• if the limit doesn't exist there,
• or if the limit and the function value aren't the same there

Continuity is when the limit as x approaches p of a function is the same as the function value at p.

Polynomials are functions which are continuous everywhere.

The idea of continuity is that you can draw the function without picking up your pencil.  In other words the function doesn't have a gap or a jump at the point in question.

Continuity is when the limit as x approaches p of a function is the same as the function value at p.

Intuitively it means you can draw it without picking up your pencil.  In other words the limit as x approaches p of a function is the same as the function value at p.

Final Continuity Quiz

Question

If a function isn't defined at a point p, can it be continuous at p?

No.  A function must be defined at the the point to be continuous there.

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Question

If the limit from the left at p of a function isn't equal to the limit from the right at p, can the function be continuous at p?

No.  The limit from the left and the limit from the right at p need to have the same value.

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Question

If the function value at p isn't the same as the limit as x approaches p, can the function be continuous at p?

No.  The limit of the function as x approaches p has to be the same as the function value at p for it to be continuous.

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Question

What are the 3 steps to checking to see if a function is continuous at a point?

Step 1:  make sure the function is defined at the point.

Step 2:  make sure the limit as x approaches the point of the function exists.

Step 3:  make sure the limit and the function value are the same thing.

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Question

If a function is not continuous at p, what is it called?

Discontinuous, or you could say that the function has a discontinuity at p.

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Question

True or False:  If a function has a limit as x approaches p, it can still fail to be continuous as x approaches p.

True.  If the function value at p is not the same as the limit as x approaches p, then it will still fail to be continuous.

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Question

Can a function that is continuous at p have a removable discontinuity at p?

No.  For there to be a removable discontinuity at p first there has to be a discontinuity.

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Question

True or False:  A function is either continuous or has a removable point of discontinuity.

False:  The function may have a non-removable point of discontinuity.  An example of this happening is when a function has a vertical asymptote.

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Question

How do you tell if a discontinuity at p is removable or not?

If the limit of the function as x approaches p exists, BUT f(p) is not defined, then it is a removable discontinuity.  Otherwise, it is a non-removable discontinuity.

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Question

How can you tell by looking at the graph of a function that it has a removable discontinuity at a point?

There is a hole in the graph at that point.

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Question

How can you tell by looking at the graph of a function that it has a non-removable discontinuity?

It has a vertical asymptote.  Anywhere a vertical asymptote happens is a non-removable point of discontinuity. It can also "jump" from one value to the next, known as a jump discontinuity.

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Question

Can a function have both a jump discontinuity and an infinite discontinuity at the same point?

No.  For it to be a jump discontinuity the limit from the left and the limit from the right at that point have to be numbers.  For it to be an infinite discontinuity one of those limits needs to be infinity, which is not a number.

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Question

Can a function have both a jump discontinuity and a removable discontinuity at the same point?

No.  For it to be a jump discontinuity the limit from the left at that point has to be different from the limit from the right at that point, but a removable discontinuity needs them to have the same value.

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Question

Are jump discontinuities removable discontinuities?

No.  They are non-removable discontinuities.

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Question

Give an example of a function with a jump discontinuity.

There are lots of them, but the Heaviside function, also known as the unit step function, is one of the more famous ones.

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Question

Can a function have both a jump discontinuity and an infinite discontinuity?

Yes, but not at the same point.

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Question

Can a function have both a jump discontinuity and a removable discontinuity?

Yes, but not at the same point.

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Question

State the definition of a function being continuous on its domain.

A function is continuous on its domain if it is continuous at every point in its domain.

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Question

What are the intervals of continuity of a function?

They are the intervals in the function's domain where the function is continuous.

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Question

Where are rational functions continuous?

Rational functions are continuous on their entire domain.

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Question

True or False:  The Sum, Difference, Product, Quotient, and Constant Multiple properties all hold for discontinuous functions.

False.  You can find examples of discontinuous functions where all of these properties fail.

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