AP Calculus AB and BC glossary
Continuity on an interval
A function is continuous on a closed interval when it is continuous at every interior point and one-sidedly continuous at each endpoint. This is the exact hypothesis required by the Extreme Value Theorem, the Intermediate Value Theorem, and the Mean Value Theorem.
Endpoints only get a one-sided requirement, because there is nothing on the far side to approach from. That is why a closed interval works even though the two-sided limit at an endpoint does not exist.
The mistake
Quoting a theorem without checking its interval hypothesis. Every major existence theorem in the course fails without continuity on a closed interval, and stating that hypothesis is worth a point on free response.
Appears in: Unit 1: Limits and Continuity