Part of learning calculus is being able to figure out a function’s discontinuity. That means you have to be able to tell if a function is continuous or discontinuous. This page explains how to do just that! It first goes over the types. Then it explains how to find if the function has one of those types in four general steps.
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Types of Discontinuity
No discontinuity: The function is continuous. In this case, all the limits exist.
Removable discontinuity: The function is not continuous because there is a hole. In this case, all limits exist.
Nonremovable discontiuity: The function is not continuous because there is a jump or because there is an infinite portion which leads to an asymptote. If it is jump discontinuity, then one sided limits exist at the place of the jump. If it is infinite discontinuity, then no limits exist at the place of the asymptote.
These images came from a San Francisco University web page.
Step 1: Is it a step function?
You can recognize a step function by the double brackets [[ ]] or single brackets [ ].
The example is the step function y = [x]. As you can see, it is not continuous because there are breaks between each step.
Yes, it is a step function means it is discontinuous. Specifically it has jump or nonremovable discontinuity.
No, it is not a step function. Proceed to Step 2.
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Step 2: Is it a fraction with a variable in the denominator?
Yes, there is a variable in the denominator, then there is likely discontinuity. However, there are still more steps you need to take to find out. Proceed to Step 2A.
If there is not a variable in the denominator, proceed to Step 3.
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Step 2A: Can the denominator be factored?
No, the denominator cannot be factored, then the function has nonremovable discontinuity because there is an asymptote when the denominator is 0.
In the example, there is an asymptote at x = 5/2. That is because the denominator is 0 when x is 5/2. Therefore, the function has nonremovable discontinuity at x = 5/2.
Yes, the denominator can be factored. Proceed to Step 2B.
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Step 2B: Factor the whole fraction.
Factor both the denominator and the numerator.
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Step 2C: Find the terms that can be crossed out.
Crossed out term: x – 2
Not crossed out term: x + 5
Terms can be crossed out if they are in the numerator and denominator. Once you’ve found the crossed out terms, set them equal to 0. The function is undefined at those points. Therefore, there are holes creating removable discontinuity at those points.
For the example, do x – 2 = 0. So there is a hole when x = 2. There is removable discontinuity at x = 2.
Proceed to Step 2D to find out of there are any asymptotes.
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Step 2D: Look at the terms in the denominator that cannot be crossed.
If there is a time when these terms equal 0, then there are asymptotes creating infinite or nonremovable discontinuity.
In the example, x + 5 cannot be crossed out. Since x + 5 = 0 when x = -5, there is an asymptote at x = -5. Therefore, the function is discontinuous at x = -5.
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Sometimes the term that cannot be crossed out never equals 0. This means that the function is continuous (unless there was discontinuity because of other terms).
In the example, the only term in the denominator cannot equal 0. Therefore the function is continuous.
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Step 3: Is it a piecewise function?
Yes, it is a piecewise function. Then you’re going to have to graph it. Just draw a small grid on your paper and sketch the graph. I know you avoid graphing like the plague, but it’s really the easiest and most accurate way to find out if there is any discontinuity.
No, it is not a piecewise function. Proceed to Step 4.
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Step 3A: Do all the pieces touch?
Yes, all the pieces touch. Then the graph is continuous. The example above shows a continuous piecewise function.
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Yes, except for one hole. Then the graph has hole or removable discontinuity.
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No, not all of the pieces touch. Then the graph has jump or nonremovable discontinuity.
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Step 4: Is it a tan, csc, sec, or cot function?
Yes, it is. Then the function has infinite or nonremovable discontinuity.
No, it is not. Then the function is continuous!
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Other Math Resources
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