Removable Discontinuities: Definition & Concept
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- Author Cynthia Helzner
Cynthia Helzner has tutored middle school through college-level math and science for over 20 years. She has a B.S. in microbiology from The Schreyer Honors College at Penn State and a J.D. from the Dickinson School of Law. She also taught math and test prep classes and volunteered as a MathCounts assistant coach.
View bio - Instructor Yuanxin (Amy) Yang Alcocer
Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
View bio - Expert Contributor Kathryn Boddie
Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.
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Table of Contents
- What is a Removable Discontinuity?
- Removable vs Non-Removable Discontinuity
- How to Find Removable Discontinuity
- Removable Discontinuity Graph
- Lesson Summary
- FAQs
- Activities
Additional Examples
In the following examples, students will identify if a function has a removable discontinuity, both graphically and algebraically. In particular, the problems using a graph will emphasize the visual differences between removable discontinuities and other discontinuities, such as vertical asymptotes.
Examples
1) Does the function graphed below have a removable discontinuity? If so, where does it occur?
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2) Does the function graphed below have a removable discontinuity? If so, where does it occur?
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3) Does the function below have a removable discontinuity? If yes, how could you redefine the function so it does not have this discontinuity?
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4) Fully factor the rational function to find if it has any holes.
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Solutions
1) The function has a discontinuity, but it is not a removable discontinuity. There is a vertical asymptote at x = 2.
2) The function has a removable discontinuity at x = - 3. We know this is a removable discontinuity because, when graphed, it appears as a hole.
3) Yes, the function has a removable discontinuity since f(2) = 5, but if we substitute x=2 into f(x) = x^3 - 3x + 1 we have f(2) = 2^3 - 3(2) + 1 = 8 - 6 + 1 = 3. We could redefine the function to remove this discontinuity as:
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4) To factor the function, first factor out the greatest common factor for both the numerator and denominator. Then factor the quadratics.
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There are common factors of x and x + 1 in both the numerator and the denominator - and if we cancel the common factors out, we are no longer dividing by zero at x = 0 and x = -1. So we have holes at x = 0 and x = -1.
How do you find the removable discontinuity of a function?
If there is a common factor in the numerator and the denominator of a rational function, set that factor equal to zero and solve for x. Plug the x-value into the reduced form of the fraction to get the y-value of the hole. If there is an isolated x-value missing from the domain of a piecewise function, or the piecewise function has a piece for a single x-value that is discontinuous with its surroundings, that x-value is a removable discontinuity.
How do you know if a discontinuity is removable?
If the break in the function can be plugged with a single point, it is removable. If not, it is non-removable. Holes and point discontinuities are removable. Vertical asymptotes and jump discontinuities are non-removable.
What is a removable discontinuity?
A removable discontinuity is a break in a function that can be plugged with a single point. The two one-sided limits at x = c are equal to each other and are not infinite, but they are not equal to f(c), if f(c) exists.
Create an accountTable of Contents
- What is a Removable Discontinuity?
- Removable vs Non-Removable Discontinuity
- How to Find Removable Discontinuity
- Removable Discontinuity Graph
- Lesson Summary
What is a Removable Discontinuity?
A discontinuity is a location on a function where one would have to pick up their pencil to keep drawing the function. Discontinuities can be removable or non-removable. What is a removable discontinuity? The removable discontinuity definition is a microscopic gap in a function for which the two one-sided limits (the y-value obtained from a leftward approach and the y-value obtained from a rightward approach) are equal. In essence, a removable discontinuity could be filled with one dot from a pencil, as opposed to needing to draw a line to connect the pieces of the function to the right and left of it.
A removable discontinuity example is the hole at (1, -0.2) in the function shown in Figure 1.
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There are two ways that a removable discontinuity can be created: a common factor in the numerator and the denominator of a rational function, and a piecewise function with one piece of the function having a domain of x = c where c is a constant or a domain excluding c.
Reduceable Rational Function
A common factor in the numerator and the denominator of a rational function leads to a hole (a removable discontinuity for which the function is undefined). For example, the factor x - 1 is in the numerator and denominator of the rational function {eq}f(x) = \frac{(x-1)(x-2)}{(x-1)(x+4)} {/eq}, which is shown in Figure 1. When the fraction is reduced, that factor is reduced out, leaving {eq}g(x) = \frac{x-2}{x+4} {/eq}. It is not valid to divide by zero so f(x) is equivalent to g(x) only if x - 1 does not equal zero. Because {eq}x -1 \ne 0 {/eq}, it follows that {eq}x \ne 1 {/eq}. As a result, the graph of f(x) is the graph of g(x) but with a hole at x = 1 because the function is undefined there.
Piecewise Function
A piecewise function with a piece for x = c leads to a point discontinuity (a removable discontinuity for which the function is defined i.e., a defined blip) if {eq}\displaystyle \lim_{x \to c}f(x) \ne f(c) {/eq}. A point discontinuity looks like a hole at x = c and the point (c, f(c)) is elsewhere on the graph, as shown in Figure 2.
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Alternatively, if c is not in the domain but the numbers around it are, there will be a hole. For example, {eq}\begin{equation*} h(x) = \left\{ \begin{array}{ll} -x & \quad x < 0 \\ x & \quad x > 0 \end{array} \right. \end{equation*} {/eq}, which is shown in Figure 3.
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Non-Removable Discontinuity
A non-removable discontinuity is a break in a function that cannot be plugged with a single point. In other words, a non-removable discontinuity is an x-value for which the two one-sided limits are either not equal or they are equal and infinite. For example, a vertical asymptote (a vertical line that the graph approaches but never hits because the function is undefined at that x-value) is non-removable because it cannot be plugged with a single point. See Figure 1 for an example of a vertical asymptote at x = -4.
The other type of non-removable discontinuity is a jump discontinuity, which is shown in Figure 4. A jump discontinuity is a vertical gap in a function.
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Yes! Keep playing. Your next lesson will play in 10 seconds- 0:00 Removable…
- 0:40 Created Discontinuity
- 1:56 What Are Holes?
- 3:11 Finding Holes
- 4:00 Lesson Summary
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Removable vs Non-Removable Discontinuity
To identify a removable vs. non-removable discontinuity, see if the break in the function can be plugged with a single point. If yes, it is removable; if no, it is non-removable. One of each is shown in Figure 5. Be careful not to rely solely on whether the two one-sided limits are equal. The one-sided limits at the asymptote in Figure 5 are both {eq}-\infty {/eq} and yet it is non-removable because it cannot be plugged with a single point.
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How to Find Removable Discontinuity
Here are the steps for how to find removable discontinuity values for a rational function:
- Factor the numerator and denominator.
- Identify any factors that the numerator and denominator have in common.
- Set each of those factors equal to zero and solve for x.
- Plug the x-value(s) into the reduced form of the fraction to get the y-value of each hole.
Example (rational function)
Find the removable discontinuity for {eq}p(x)=\frac{x^2 - 6x-7}{x^3 - 2x^2 - x + 2)} {/eq}.
Factoring the numerator and denominator yields {eq}p(x)=\frac{(x+1) (x-7)}{(x-2) (x+1) (x-1)} {/eq}. The numerator and denominator have (x + 1) in common.
x + 1 = 0
x = -1
The reduced form of the fraction is {eq}\frac{x-7}{(x-2)(x-1)} {/eq}. Plugging in x = -1 yields {eq}y = \frac{-1-7}{(-1-2)(-1-1)} = \frac{-8}{6} = -\frac{4}{3} {/eq}. Thus, the hole is {eq}(-1, -\frac{4}{3} ) {/eq}.
Here are the steps for how to find removable discontinuities for a piecewise function:
- See if there is an isolated x-value missing from the domain of a piecewise function and the two one-sided limits at that x-value are equal and finite. If so, there is a removable discontinuity at that x-value.
- See if the piecewise function has a point for a single x-value that is discontinuous with its surroundings (i.e., the two one-sided limits at that x-value are equal and finite but are not equal to f(x) at that point). If so, there is a removable discontinuity at that x-value.
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Removable Discontinuity Graph
Turning a continuous graph into a removable discontinuity graph is simply a matter of multiplying the function by a fraction composed of a factor over the same factor e.g. {eq}\frac{x+2}{x+2} {/eq}. For example, Figure 6 shows the graph of {eq}y = x(x+5) {/eq} and Figure 7 shows the graph of {eq}y = \frac{x(x+5)(x+2)}{x+2} {/eq}.
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A hole is typically invisible on a graphing calculator but it can be drawn in by hand or labeled with some graphing programs. A small, empty circle on a function means a removable discontinuity. A hole in a graph is an isolated x-value at which the two one-sided limits are equal and finite but the function is undefined at that x-value. For example, Figure 7 shows a parabola with a hole when x = -2 because the function is undefined at that x-value. So, finding removable discontinuity values from a graph is simply a matter of finding the labeled holes. The steps for how to find removable discontinuity values algebraically from a function are described earlier in this article.
Removable Discontinuity Limit
A removable discontinuity is an isolated x-value, c, at which the limit of the function exists (i.e., the two one-sided limits are equal) and is finite but the function is undefined at c or the removable discontinuity limit is not equal to f(c). For example, Figure 7 has a hole at (-2 -6) because the limit of the function at x = -2 exists (it is -6) but the function is undefined at x = -2 as a result of x = -2 making the function's denominator equal to zero. A point discontinuity at x = 0 is shown in Figure 2 because the limit of the function at x = 0 exists (it is 0) but it is unequal to f(0), which is 0.4.
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Lesson Summary
A discontinuity is a gap in a function. A removable discontinuity is an x-value in a function for which the two one-sided limits are equal and finite but either the function is undefined at c or the limit at c is not equal to f(c). The former produces a hole; the latter produces a point discontinuity (i.e., a defined blip). A hole can be found in a rational function or a piecewise function, while a point discontinuity is only found in a piecewise function. A removable discontinuity can be plugged with a single point. In contrast, a non-removable discontinuity is a break in a function that cannot be plugged with a single point. In other words, a non-removable discontinuity is an x-value for which the two one-sided limits are either not equal or they are equal and infinite. The former produces a vertical asymptote (the function is undefined at the asymptote's x-value) or a jump discontinuity; the latter only produces a vertical asymptote. A function can have any number of any of those types of discontinuity.
To find the removable discontinuities of a rational function, factor the numerator and denominator, set any common factors equal to zero, and solve for x. Plug those x-value(s) into the reduced fraction to get the y-value(s) of the hole(s). If there is an isolated x-value missing from the domain of a piecewise function or the piecewise function has a point that is discontinuous with its surroundings, that x-value is a removable discontinuity.
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Video Transcript
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 in the graph at that location. A removable discontinuity is marked by an open circle on a graph at the point where the graph is undefined or is a different value, like this:
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Do you see it? There is a small open circle at the point where x ≈ 2.5.
There are two ways a removable discontinuity can be created. Let's talk about the first one now.
Created Discontinuity
A removable discontinuity can be created by defining a blip in the graph like this.
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This 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 that 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.
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If we were to graph this equation, we 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 is 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. This factor can be canceled out but needs to still be considered when evaluating the function, such as when graphing or finding the range. When dealing with a function like this, there will be some point where the function is undefined. Look at this function, for example.
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This function has the factor x - 4 in both the numerator and denominator. What happens at the point x = 4? Let's see.
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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. When you do so, the function can be rewritten as follows.
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Having rewritten our function, we see that the function generally looks like the graph of y = x except at the point x = 4.
Finding Holes
Holes can exist for rational expressions when you have factors that cancel in the numerator and denominator. Perhaps you can factor a polynomial in either the numerator or denominator or both. To find holes, recall that there must be a common factor in the top and bottom of the fraction. Let's see how this process works for a sample function.
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After factoring my function, we have found that there is a common factor of x + 2 in the numerator and denominator. Solving that for 0, there is a hole at x = -2. When you graph what is left, you get a line with a small open circle at x = -2.
There can be multiple holes in a function. If you have a polynomial in the denominator, there may be more than one hole in the function. For example, in this function there are two places where it is undefined:
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In this rational function, the x + 1 cancels out but still must be considered when graphing the function. There will be two places where this function is undefined, x = -1 and x = -2. At x = -1 there will be a hole. The factor x + 2 does not cancel out but -2 becomes an asymptote (a vertical line denoting a value that the graph approaches but never reaches). This will be covered in other lessons.
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 by defining a blip in the function and the other way is by the function having 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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