Removable And Non Removable Discontinuities

In the Continuity article, we learned three criteria needed for a function to be continuous. Think that all 3 of these criteria must be met for continuity at a betoken. Let's consider the 3rd criterion for a minute "the limit as x approaches a bespeak must be equal to the function value at that point". What if, say, this is not met (only the limit still exists)? What would that look like? We call it a removable discontinuity (also known as a hole)! Allow's take a farther expect.

Removable Signal of Discontinuity

Let'southward go dorsum to the scenario in the introduction. What happens if the limit exists, but isn't equal to the part value? Call back, that by saying the limit exists what you actually are maxim is that it is a number, not infinity.

If a function is not continuous at , and

exists, and then we say the function has a removable aperture at .

Hither, we ascertain as a removable point of aperture.

Ok, that'southward groovy, but what does a removable discontinuity look like? Consider the image below.

Example of a function with a removable discontinuity at x = p | StudySmarter Original

In this epitome, the graph has a removable discontinuity (aka. a pigsty) in it and the function value at is instead of the you would demand it to be if you wanted the part to exist continuous. If instead that hole were filled in with the point higher up it, and the point floating there removed, the function would go continuous at . This is called a removable aperture.

Removable Discontinuity Example

Let'south take a look at a few functions and make up one's mind if they have removable discontinuities.

Removable Discontinuity Graph

Does the function have a removable discontinuity at ?

Answer:

Outset, notice that the function isn't defined at , then it isn't continuous there. If the role is continuous at , and then it certainly doesn't have a removable discontinuity there! So at present y'all need to cheque the limit:

Since the limit of the function does exist, the discontinuity at is a removable discontinuity. Graphing the function gives:

Removable discontinuity graph example StudySmarter This function has a pigsty at x=3 considering the limit exists, notwithstanding, f(3) does not exist, StudySmarter Originals

So you can see in that location is a hole in the graph.

Non-removable Discontinuities

If some discontinuities can be removed, what does it mean to be not-removable? Looking at the definition of a removable aperture, the part that can become wrong is the limit not existing. Non-removable discontinuities refer to two other chief types of discontinues; jump discontinuities and infinite/asymptotic discontinuities. You tin larn more than about them in Jump Discontinuity and Continuity Over an Interval.

Non-removable Discontinuity Graph

Looking at the graph of the piecewise-defined function beneath, does it have a removable or not-removable signal of discontinuity at ? If it is non-removable, is it an space discontinuity?

Removable discontinuity example infinite discontinuity StudySmarter Function with a not-removable discontinuity | StudySmarter Original

Respond:

From looking at the graph you can run into that

and that

which means the function is not continuous at . In fact, it has a vertical asymptote at . Since those two limits aren't the aforementioned number, the function has a non-removable discontinuity at . Since one of those limits is space, you lot know information technology has an infinite aperture at .

Deciding if the office has a removable or non-removable point of discontinuity

Removable Aperture Limit

How can you lot tell if the discontinuity of a function is removable or not-removable? Just expect at the limit!

If the limit from the left at p and the right at p are the same number, but that isn't the value of the function at p or the function doesn't accept a value at p, and so at that place is a removable aperture.
If the limit from the left at p, or the limit from the correct at p, is infinite, then at that place is a non-removable point of discontinuity, and it is chosen an infinite discontinuity.

What kind of discontinuity, if any, does the part in the graph have at p?

Removable discontinuity example graph StudySmarter This role has a removable aperture at x=p considering the limit is defined, however, f(p) does not exist | StudySmarter Originals

Reply:

Y'all tin encounter looking at the graph that the function isn't fifty-fifty defined at p. However the limit from the left at p and the limit from the correct at p are the same, then the role has a removable indicate of discontinuity at p. Intuitively, it has a removable discontinuity because if you just filled in the hole in the graph, the function would be continuous at p. In other words, removing the aperture ways changing just one point on the graph.

What kind of discontinuity, if any, does the role in the graph have at p?

Removable discontinuity example graph StudySmarter This role is defined everywhere | StudySmarter Originals

Unlike in the previous example, you tin can run across looking at the graph that the function is defined at p. All the same the limit from the left at p and the limit from the right at p are the same, so the function has a removable signal of discontinuity at p. Intuitively, it has a removable discontinuity because if you only changed the role and then that rather than having it filled in the hole, the office would be continuous at p.

Looking at the graph of the piecewise-defined part below, does it have a removable, not-removable discontinuity, or neither of the two?

Removable Discontinuity graph non-removable jump StudySmarter Graph of a part with a discontinuity at 10=two, StudySmarter Original

Answer:

This office is clearly not continuous at 2 because the limit from the left at 2 is not the same as the limit from the right at two. In fact

and

And so we know that

the limit from the left at 2 and the limit from the right of 2 don't take the same value
the limit from the left isn't infinite, and the limit from the right isn't infinite at 2 either,

Therefore, this function has a non-removable aperture at 2, however, it is not an infinite discontinuity.

In the example above, the role has a leap discontinuity at . For more information on when this happens, run into Leap Aperture

Looking at the graph below, does the part have a removable or non-removable point of aperture at ?

Removable Discontinuity infinite type graph StudySmarter Graph of a role with a discontinuity at x = two, StudySmarter Original

Respond:

This function has a vertical asymptote at . In fact

and

And then this function has a non-removable betoken of discontinuity. It is called an infinite discontinuity because ane of the limits is infinite.

Removable Discontinuity - Key takeaways

If a office is not continuous at a point, we say "it has a indicate of discontinuity at this bespeak".
If a function is not continuous at a point, then we say the role has a removable discontinuity at this betoken if the limit at this bespeak exists.
If the function has a removable discontinuity at a indicate, and then is called a removable point of discontinuity (or a hole).