MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function f is defined on the closed interval [−4,4]. Its graph is described below.
• For −4≤x<−2, the graph is the line segment joining (−4,1) and (−2,3).
• At x=−2, there is an open circle at the point (−2,3) and a solid point at (−2,1).
• For −2<x<1, the graph follows the curve y=21(x+2)2+1.
• At x=1, the graph has a solid point at (1,4).
• For 1<x<3, the graph is the horizontal line y=4.
• At x=3, there is an open circle at (3,4) and no other point plotted.
• For 3<x≤4, the graph follows the curve y=x−26.
Answer each of the following. Use precise limit notation and clearly communicate your reasoning.
(1) Identify all real numbers c in [−4,4] at which f is not continuous.
(2) For each value of c found in part (a), classify the discontinuity at x=c as removable, jump, or infinite.
(3) Provide a justification for each classification in part (b) by explicitly evaluating the appropriate one-sided limits and discussing whether f(c) is defined.