Lesson 10: 1-10 Types of Discontinuities
Duration of Days: 1
Lesson Objective
The student will be able to:
Classify discontinuities into three main types: Removable, Jump, and Infinite.
Identify the algebraic cause of each type (e.g., common factors vs. non-zero constants over zero).
Describe the behavior of one-sided limits at each type of discontinuity.
Determine if a discontinuity is "removable" by redefining a single point.
Why is a "hole" called a removable discontinuity, while a "jump" is not?
What happens to the denominator of a rational function at a vertical asymptote?
How can you look at a piecewise function's equations and predict if there will be a jump at the boundary?
Removable Discontinuity (Hole)
Non-Removable Discontinuity
Jump Discontinuity
Infinite Discontinuity (Vertical Asymptote)
Essential Discontinuity
One-Sided Convergence
F-IF.A.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.B.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.
F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.9: Compare properties of two functions each represented in a different way
Description: This lesson provides a taxonomy of "graphical failures.
"Removable: The limit exists, but the point is missing or displaced (0/0 form).
Jump: The left-hand and right-hand limits exist but are not equal (common in piecewise functions).
Infinite: The function approaches infinity.
Purpose: Understanding these types allows students to communicate precisely about function behavior. On the AP Exam, students are often asked to "describe the symmetry or discontinuities" of a function. It also prepares them for L'Hôpital's Rule and Improper Integrals later in the course.6.
DOK Levels
DOK Level 1 (Recall): Naming the type of discontinuity shown on a provided graph.
DOK Level 2 (Skill/Concept): Analyzing a rational function's equation to find the x-values of holes and asymptotes.
DOK Level 3 (Strategic Thinking): Creating a function that possesses at least two different types of discontinuities simultaneously.
For Struggling Learners (Scaffolding):
The "Fix-It" Test: Ask, "Can I fix this graph by drawing just one tiny dot?" If yes, it's Removable. If no, it's Non-Removable.
For Advanced Learners (Extension):
Oscillating Discontinuity: Revisit F(x) = sin(1/x) and discuss why this is called an "Essential" discontinuity (it doesn't fit the standard three).
College Board AP Classroom Assessments