The graph of a piecewise function has the following features
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
The graph of a piecewise function p has the following features.
• On x<−2, the graph is the line with equation y=21x+2.
• At x=−2, the line reaches the point (−2,1), but that point is shown as an open circle.
• For −2<x<1, the graph follows a smooth cubic curve that approaches (−2,1) from the right and approaches (1,−1) from the left.
• At x=1, the point (1,−1) is an open circle.
• For 1<x<3, the graph is the horizontal line y=−1.
• At x=3, there is a filled dot at (3,2) while the horizontal line y=−1 continues but now as an open circle at (3,−1).
• For x>3, the graph follows the ray beginning at (3,2) with slope 3.
Respond to the prompts below.
(a) List the coordinates of all points that must be added to the graph in order to remove every discontinuity that can be removed without altering the shape of any existing curve segment.
(b) Construct a precise, piecewise definition of a function P that includes the additional points from part (a) and coincides with the original graph elsewhere. Clearly indicate the domain of each piece, including whether endpoints are open or closed.
(c) Justify that the function P you have defined is continuous at every x where a point was added, and explain why no further adjustments can render P continuous at x=3.