Exploring Types of Discontinuities
Identify the actual obstruction to continuity and decide whether one replacement value can repair it.
College Board Topic 1.10 · CED p. 42 (PDF p. 47) · CED effective Fall 2020; project course outline 2026–27.
A discontinuity means the function fails to connect its actual value with its nearby behavior. A removable discontinuity has a finite common limit but a missing or wrong target value. A jump has distinct finite one-sided limits. An infinite discontinuity involves unbounded behavior near the target.
Compare (x²−1)/(x−1) at 1, a step function that changes from 0 to 1 at 0, and 1/(x−2) at 2. The first has a hole that one value can repair; the second has a jump that a single value cannot repair; the third has a vertical asymptote that a finite point cannot repair.
Not every discontinuity belongs to those three simple pictures. Oscillation such as sin(1/x) near 0 also prevents continuity. Describe the actual obstruction rather than forcing an unsuitable label.
Let's work through it
- For f(x)=0 when x<0 and f(x)=1 when x≥0, compute left limit 0 and right limit 1. Reason: each side follows a constant rule.
- The two-sided limit does not exist. Reason: the finite one-sided limits disagree.
- Classify a jump at 0. Reason: changing f(0) cannot make the two traces meet.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
The limit is 7 but f(c)=2. What type of discontinuity is this?
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Retrieval 1
The limit is 7 but f(c)=2. What type of discontinuity is this?
Worked answer and reason
Removable.
Replacing the target value by 7 would match the common nearby limit.
Retrieval 2
Can assigning f(0)=100 repair 1/x at 0?
Worked answer and reason
No.
Its unbounded one-sided behavior has no finite common limit.
Catch the tempting mistake
- Any missing point can be filled to make a function continuous.
- A finite common limit is needed; jumps and unbounded behavior cannot be repaired by one finite value.
Calculator workspace and limits
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Scientific calculator
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Put it to work
- Unit 1 problem 16 · No calculator · analytic
For the teacher
- Use a classification task that includes an oscillatory counterexample.
- Ask what information is required before calling a discontinuity removable.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Discontinuity
A discontinuity is a failure of continuity at an input being considered. The obstruction can be a missing or mismatched value, unequal one-sided limits, oscillation, or unbounded behavior. For example, assigning f(0)=5 while f(x)=x elsewhere creates a discontinuity at 0. Identifying the obstruction determines whether changing one value can repair it.
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Removable discontinuity
A removable discontinuity occurs when a finite two-sided limit exists but the function is missing or has a different value at the target. Changing or defining that single value to equal the limit repairs continuity. For example, (x²−1)/(x−1) has a removable discontinuity at 1 and can be repaired by setting its value there to 2. A jump with unequal one-sided limits cannot be repaired this way.
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Vertical asymptote
A vertical asymptote x=c occurs when function values tend to positive or negative infinity as x approaches c from at least one side. Being unbounded near c is insufficient if the values oscillate without tending to either signed infinity. For example, 1/x has vertical asymptote x=0, tending to positive infinity from the right and negative infinity from the left. A zero denominator alone does not prove an asymptote because cancellation can instead leave a removable hole.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.