AP Calculus AB and BC glossary
Oscillating discontinuity
An oscillating discontinuity happens where a function swings between values infinitely often as it approaches a point, so no limit exists. The standard example is sine of one over x at x equals zero, which crosses every value between negative one and one infinitely many times.
It is a fourth kind of failure alongside removable, jump, and infinite. The function stays bounded, so nothing runs to infinity, and yet the values never settle, so does not exist.
The mistake
Assuming a bounded function must have a limit. Boundedness is not enough; the values also have to converge.
Appears in: Unit 1: Limits and Continuity