Calculus I · Limits and Continuity · lesson
Types of Discontinuity
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 |
Read this graph as text
Four types of discontinuity. A two-by-two gallery compares behavior at x = 0. Removable shows y = x + 1 with an open point at (0, 1). Jump shows y = 0 to the left and y = 2 to the right with open and filled markers. Infinite shows y = 1/x on split domains with a vertical asymptote. Oscillatory shows y = sin(1/x) on split domains that never settle near zero.
Panel titles name each type. Open and filled markers distinguish inclusion, the jump branches use different line styles, and the infinite panel has a dashed asymptote.
Why it matters: Compare removable, jump, infinite, and oscillatory discontinuities in a consistent four-panel layout.
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.
After the explanation
Use the section idea
Do the limit, the function value, and the surrounding domain fit together at the point or across the interval?
Continuity is a three-part agreement: the value exists, the two-sided limit exists, and those two quantities are equal.
At a point, test the three conditions in order; on an interval, check the domain and endpoints before invoking any continuity theorem.
A sign change supports the Intermediate Value Theorem only when continuity holds on the entire closed interval, and it does not prove uniqueness.
You are ready to continue when you can classify a break, decide whether one value can repair it, and state every IVT hypothesis aloud.
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