Calculus I · Limits and Continuity · lesson
Types of Discontinuity
Learn Types of Discontinuity with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
Where this chapter fits
Chapter 5: Continuity
Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.
Reading lens: Do the limit, the function value, and the surrounding domain fit together at the point or across the interval? Keep that question in view while reading Types of Discontinuity; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Which Functions Are Continuous? to How to Repair a Removable Discontinuity. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Distinguish removable, jump, infinite, and oscillatory discontinuities; connect each classification to the relevant limit behavior.
Classifying Discontinuities
| Type | Limit behavior | Typical graph feature |
|---|---|---|
| Removable | Finite two-sided limit exists, but value is missing or wrong | Hole or displaced dot |
| Jump | Finite one-sided limits exist but disagree | Sudden step |
| Infinite | Function becomes unbounded | Vertical asymptote |
| Oscillatory | Outputs do not settle to one value | Endless oscillation |
The four principal discontinuity types encountered in first-semester calculus.
Classify a rational function's discontinuities
Classify every discontinuity of
Show worked solution
Factor numerator and denominator:
For ,
At , the common factor cancelled. The simplified function is finite:
Thus is a removable discontinuity with a hole at .
At , the remaining denominator is zero and the numerator is nonzero. Thus is an infinite discontinuity and vertical asymptote.
Source & rights
Original instruction with traceable references.
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