Determine locations of removable discontinuities by graphical, numeric, or analytic methods
Determine when and how discontinuous functions can be made continuous
I can locate removable discontinuities by using the definitions of limits and continuity.
I can calculate the needed function value to retain a limit and create continuity.
I can use extended functions to define or redefine the y-value at a point to match the limit at that point.
I can use the definition of continuity to justify my solutions.
Quick Lesson Plan
This lesson follows the adventures of Pia and Pepe at Starbucks. Previously, we examined types of discontinuities. Today, students define points, if possible, to remove a discontinuity where a limit exists.
Assign work with piecewise functions often to highlight domain restrictions, graphing techniques, and behavior of the function at important domain values. Continue to require demonstration of the definition of continuity, especially when using analytic methods.
Remind students that the AP Exam will ask them to evaluate function behavior at “interesting” values of the domain. In the practice exam question here, x = 2 is the chosen value, of course!
After following rigid mathematical rules for years, many of us (teachers included!) are troubled by the idea that we can reassign given values of a function or simply calculate a function value to create continuity. When this is requested, however, encourage students to check their calculations with both definitions to confirm that the value they have found equals the limit from each side.