Lesson 9: 1-9 Multiple Representations
Duration of Days: 1
Lesson Objective
The student will be able to:
Define continuity at a point using the three-part formal definition.
Determine if a function is continuous at a specific x-value by checking the existence of the limit, the existence of the function value, and their equality.
Identify points of discontinuity from a graph or an algebraic expression.
Justify why a function is or is not continuous using the formal three-step criteria.
What is the difference between saying a limit exists and saying a function is continuous?
Can a function be continuous at a point if it is undefined at that point?
Why do all three conditions of the continuity definition have to be met? (What happens if only two are met?)
Continuity at a Point
The Three-Part Definition
Discontinuity
Point of Discontinuity
Removable vs. Non-removable
Interval Continuity
MA.F-IF.C.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Description: This lesson formalizes the "pencil test" Students learn that for f(x) to be continuous at x=c:f(c) must exist
Purpose: This is one of the most frequently tested concepts on the AP Exam. It is the prerequisite for almost every major theorem in Calculus, including the Intermediate Value Theorem (1.16), the Mean Value Theorem (5.1), and the Extreme Value Theorem (5.2).6.
DOK Levels
DOK Level 2 (Skill/Concept): Identifying points of discontinuity in a piecewise function.
DOK Level 3 (Strategic Thinking): Writing a formal justification for continuity using the three-part definition (essential for FRQ points).
For Struggling Learners (Scaffolding):
The Continuity Checklist: Provide a 3-box checklist for every problem. If they can't check all three, it's not continuous.
Advanced Learners:
Dirichlet Function: Introduce a "pathological" function that is discontinuous everywhere to challenge their intuition.
AP College Board Assessments