Lesson 11: 1-11 Defining Continuity at a Point
Duration of Days: 2
Lesson Objective
The student will be able to:
Apply the three-part definition of continuity to provide a formal mathematical justification.
Determine the value of a constant (k or a) that makes a piecewise function continuous at a boundary.
Communicate reasoning using proper limit notation and logical sequencing.
Verify continuity for various function types, including trigonometric and rational functions.
Why is "I can draw it without lifting my pencil" not an acceptable justification on a formal exam?
In a piecewise function, how do we mathematically "force" the two pieces to meet at the same point?
If a function is continuous on the interval (a, b), what must be true for it to be continuous on the closed interval [a, b]?
Formal Justification
Existence
Equality of Limit and Value
Boundary Point
Piecewise Continuity
Closed vs. Open Interval Continuity
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Description: This lesson focuses on the logic of continuity. Students are often given piecewise functions where one piece is a function of x and k. They must set the left-hand limit equal to the right-hand limit to find the value of k that "glues" the function together.
Purpose: This lesson transitions students from "knowing" to "proving." The three-part test is a template for mathematical communication. It also introduces the concept of one-sided continuity at endpoints, which is necessary for understanding the Intermediate Value Theorem in the next lesson.
DOK Levels
DOK Level 2 (Skill/Concept): Solving for a missing constant k in a piecewise function to ensure continuity.
DOK Level 3 (Strategic Thinking): Constructing a written argument that proves a function is discontinuous by identifying exactly which of the three conditions failed.
Differentiation Strategies
For Struggling Learners (Scaffolding):
Visual Match: Use a graphing calculator to show how changing the value of k physically slides one piece of a function up or down to meet the other.
For Advanced Learners (Extension):
Systems of Equations: Give a piecewise function with two unknowns (a and b) where the function must be continuous at two different boundary points, requiring a system of linear equations to solve.
AP College Board Assessments