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There are actually two ways you can get a removable discontinuity.

Learn about them in this lesson along with how to identify them and what you can do with them.

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Removable Discontinuity Defined

A removable discontinuity is a point on the graph that is undefined or does not fit the rest of the graph. There is a gap at that location when you are looking at the graph. When graphed, a removable discontinuity is marked by an open circle on the graph at the point where the graph is undefined or is a different value like this.

A removable discontinuity.
Defining a blip.

removable discontinuity

The above function tells us that the graph generally follows the function f(x)=x^2-1 except for at the point x=4. When we graph it, we will need to draw a little open circle at the point on the graph and mark that it equals 2 at that point.

This is a created discontinuity. If you were the one defining the function, you can easily remove the discontinuity by redefining the function. Looking at the function f(x)=x^2-1, we can calculate that at x=4, f(x)=15. So, if we redefine our point at x=4 to equal 15, we will have removed our discontinuity.

Redefining the blip so it goes away.
removable discontinuity

If we were to graph the above, we would get a continuous graph without any discontinuities. When you see functions written out like that, be sure to check whether the function really has a discontinuity or not.

Sometimes the function is continuous but just written like it isn’t just to be tricky.

What Are Holes?

Another way we can get a removable discontinuity is when the function has a hole. A hole is created when the function has the same factor in both the numerator and denominator.

When you get a function like that you will get into a situation at some point where the function is undefined. Look at this function, for example.

A function with a hole.
removable discontinuity

This function has the factor x-4 in both the numerator and denominator.

What happens at the point x=4? Let’s see.

The function is undefined at x=4.
removable discontinuity

We get an interesting answer of 0/0 which in mathematical terms is undefined. So this function is undefined at the point where x=4. We have a removable discontinuity here because the function has a hole at x=4 caused by having the same factor in both the numerator and denominator.

We can redefine our function to account for this hole by recalling that if you have the same factor in both the numerator and denominator, then you can cancel the terms. Doing so, our function can be rewritten as follows.

Redefining our function to include the hole.
Finding a hole.

removable discontinuity

After factoring my function, we have found that there is a common factor of x+2 in both the numerator and denominator. Solving that for 0, I see that I have a hole at -2.

If we have more than one common factor, we may have more than one hole in the function.

Lesson Summary

A removable discontinuity is a point on the graph that is undefined or does not fit the rest of the graph. There are two ways a removable discontinuity is created. One way is be defining a blip in the function and the other way is by the function having a hole or a common factor in both the numerator and denominator. Removable discontinuities are marked on the graph by a little open circle.

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