What Is a Discontinuity?
A function is continuous at a point when its graph has no break, hole, or vertical asymptote there. Informally, you can think of continuity as being able to draw the graph near that point without lifting your pencil.
At a point , the function is continuous if three conditions are satisfied:
- exists.
- exists.
- The two values are equal: .
If any one of these conditions fails, the function is discontinuous at . But not all discontinuities look the same. On the AP Calculus exam, you should be able to distinguish among the major types.
The Three Main Types
Removable
The limit exists, but the function has a hole or has the wrong value at the point.
Jump
The left-hand and right-hand limits both exist but are different.
Infinite
The function grows without bound near the point, producing a vertical asymptote.
1. Removable Discontinuities
A removable discontinuity occurs when the two-sided limit exists, but the function is either undefined at the point or has a value that does not match the limit.
The classic graph is a smooth curve with a single hole.
There are actually two common ways a removable discontinuity can occur:
- The function is undefined at .
- The function is defined, but is not equal to the limit.
Consider
How to Recognize It
Look for a factor that can be canceled algebraically. If canceling the factor produces a finite value for the limit, the discontinuity is usually removable.
At a removable discontinuity, the limit can exist even though does not exist. A hole tells you something about the function's value at the point; the limit tells you what the function approaches nearby.
2. Jump Discontinuities
A jump discontinuity occurs when the function approaches one finite value from the left and a different finite value from the right.
Because the one-sided limits are different, the two-sided limit does not exist.
Suppose a piecewise function is defined by
Determine the type of discontinuity at .
Why a Jump Cannot Be Removed by Redefining One Point
For a removable discontinuity, both sides approach the same number. You can fix the hole by assigning the appropriate value to the function.
A jump is different. There is no single number you can assign to that makes both sides agree. The problem is not one missing point; the entire left-hand and right-hand behavior is different.
3. Infinite Discontinuities
An infinite discontinuity occurs when the function becomes arbitrarily large in magnitude as approaches a particular value. This produces a vertical asymptote.
Unlike a removable discontinuity or jump discontinuity, the function does not approach a finite number.
Consider
Comparing the Three Types
The fastest way to classify a discontinuity is to ask what happens to the left-hand and right-hand limits.
- Removable: both sides approach the same finite number, but the function has a hole or the wrong point value.
- Jump: both one-sided limits are finite but unequal.
- Infinite: at least one side becomes unbounded, producing a vertical asymptote.
Using a Graph to Identify the Type
On the AP exam, you may be given a graph instead of an equation. You do not necessarily need to calculate anything. Instead, focus on the behavior of the graph as approaches the suspicious point.
Look for a Hole
A single open circle on an otherwise continuous curve is the classic sign of a removable discontinuity. If the curve approaches the same height from both sides, the limit exists.
Look for a Vertical Step
If the graph approaches one height from the left and another height from the right, you have a jump discontinuity.
Look for a Vertical Asymptote
If the graph shoots upward or downward without bound near a vertical line, you have an infinite discontinuity.
An open circle often indicates a removable discontinuity, but you still need to check the behavior from both sides. The key question is whether the left- and right-hand limits approach the same finite value.
Using a Table of Values
A table can also reveal the type of discontinuity. Look at values approaching the target from both directions.
Suppose the table gives the following values near :
A Reliable AP Calculus Strategy
When asked to identify or classify a discontinuity, use this sequence every time:
- Find the point of concern. Look for a denominator of zero, a piecewise boundary, an open circle, or another obvious break.
- Check the left-hand limit. What does the function approach as ?
- Check the right-hand limit. What does the function approach as ?
- Compare the two sides. Are they the same finite number, different finite numbers, or unbounded?
- Check the actual function value. If both sides agree but the function value is missing or incorrect, the discontinuity is removable.
Worked Example: Classify Every Discontinuity
Consider the function
Classify the discontinuities at and .
Removable vs. Jump vs. Infinite
That distinction is worth memorizing. A removable discontinuity has a finite limit. A jump has two different finite one-sided limits. An infinite discontinuity involves unbounded behavior.
Common Mistakes
A hole is specifically associated with a removable discontinuity. A jump and an infinite discontinuity are different phenomena. Always inspect the one-sided behavior before assigning a type.
A function being undefined at does not automatically mean the discontinuity is infinite. For example, is undefined at , but its discontinuity is removable because the limit is finite.
For a jump discontinuity, the most important evidence is that the left-hand and right-hand limits are different. If you only calculate a value from one side, you may miss the classification entirely.
In AP Calculus, be precise. When a function grows without bound, write that the limit is infinite or does not exist as a finite number, depending on the context. The key conclusion is that there is an infinite discontinuity and a vertical asymptote.
Practice Problems
Classify the discontinuity of at .
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Suppose and . What type of discontinuity occurs at ?
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Classify the discontinuity of at .
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A graph approaches 3 from the left and 7 from the right as . What type of discontinuity occurs?
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A function has , but is undefined. Can the discontinuity be removable?
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