Lesson 12: 1-12 Confirming Continuity Over an Interval
Duration of Days: 1
Lesson Objective
The student will be able to:
Identify intervals on which a function is continuous based on its domain and points of discontinuity.
Determine continuity on a closed interval [a, b] by checking continuity on the open interval (a, b) and one-sided limits at the endpoints.
Recognize the "built-in" continuity of basic function families (polynomial, rational, radical, trigonometric, exponential, and logarithmic).
Sketch functions that meet specific continuity requirements over a given range.
How does the definition of continuity change when we look at an endpoint of a domain (like the start of a square root function)?
If two functions are continuous on an interval, are their sum, product, and composition also continuous on that interval?
Why is it important to distinguish between "continuous everywhere" and "continuous on its domain"?
Open Interval (a, b)
Closed Interval [a, b]
Continuity from the Right / Left
Endpoint Continuity
Domain of Continuity
Function Composition
F-IF.1: Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range.
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of context.
F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
Description: This lesson teaches students that continuity isn't just a point-by-point check. They learn to identify the "Domain of Continuity." but it is not "continuous everywhere." A major focus is Endpoint Continuity: for a function to be continuous on [a, b], it must be continuous at every point in between, and the limit from the right at a must equal f(a), while the limit from the left at b must equal f(b).
Purpose: Many Calculus theorems (like the Mean Value Theorem) begin with the phrase: "If f is continuous on the closed interval [a, b]..." Students must be able to verify this condition before they can use the theorem. It also prepares them for understanding Infinite Intervals and Improper Integrals.
DOK Levels
DOK Level 2 (Skill/Concept): Stating the intervals of continuity for a given rational or radical function.
DOK Level 3 (Strategic Thinking): Determining if a piecewise function is continuous on a closed interval by checking both the internal boundaries and the external endpoints.
For Struggling Learners (Scaffolding):
The "Domain First" Rule: Teach students to find the domain of the function first. Usually, the function is continuous everywhere it is defined, unless it’s a piecewise function.
For Advanced Learners (Extension):Composition Challenge: Ask students to find the interval of continuity for f(g(x)) given two functions with different discontinuities.
AP College Board Assessments