A function is represented graphically on the interval by the description below
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function f is represented graphically on the interval [−4,5] by the description below.
• For x<−3, the graph is the line segment connecting (−4,1) to (−3,3).
• At x=−3 there is a solid dot at (−3,3) and a ray that begins just to the right of x=−3 and follows the curve y=21(x+3)2+3 until it reaches x=−1.
• At x=−1 there is an open circle at (−1,5) and a solid dot at (−1,4).
• For −1<x<2, the graph follows the line of slope −1 passing through (−1,5), ending in an open circle at (2,2).
• At x=2 there is no point plotted.
• For 2<x≤5, the graph is the curve y=x−26+2.
• There is a horizontal asymptote y=2 for x>2.
Using the description above, respond to the following parts.
(a) Identify every value of x in (−4,5) at which f is not continuous.
(b) For each discontinuity in part (a), classify the type as removable, jump, or infinite. Support each classification by stating the relevant one-sided limits and function values.
(c) Write the open interval(s) contained in (−4,5) on which f is continuous. Explain how the graphical features support your answer.