AP Calculus AB and BC glossary

Piecewise function

A piecewise function uses different rules on different intervals of its domain. The interesting points are the boundaries between pieces, where you must check that the one-sided limits agree with each other and with the defined value before calling the function continuous.

Continuity at a boundary needs three things to line up: the limit from the left, the limit from the right, and the actual function value at that point. Exam questions usually ask you to solve for a constant that makes all three match.

The mistake

Checking only that the two formulas give the same value at the boundary. If differentiability is asked for as well, the derivatives of both pieces must also agree there.

Appears in: Unit 1: Limits and Continuity