Lesson 4: 2-4 Differentiability and Continuity
Duration of Days: 1
Lesson Objective
Students will be able to:
Explain the logical relationship between differentiability and continuity.
Identify the x-values where a function is not differentiable based on its graph.
Analyze piecewise functions to determine if they are differentiable at the "seam" or "break" point.
Justify the non-existence of a derivative at a point using one-sided limits of the difference quotient.
Recognize vertical tangents and cusps as locations where the slope becomes undefined.
Does the existence of a limit guarantee that a function is differentiable?
Why does continuity not automatically imply differentiability?
What specific "physical" features on a graph (corners, cusps, vertical tangents) prevent a derivative from existing?
If I know a function is differentiable at x=c, what can I conclude about its continuity at that same point?
Differentiability
Continuity
Smoothness
Corner (e.g., Absolute Value)
Cusp
Vertical Tangent
One-Sided Derivatives
Locally Linear
Piecewise Function
F-IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over an interval. Interpret average rate of change of a function in terms of an application.
This lesson explores the "One-Way Street" of Calculus: Differentiability implies Continuity. Students learn that for a derivative to exist at a point, the function must be continuous AND "smooth." The lesson focuses on identifying the four main reasons a derivative fails to exist: Discontinuities (holes, jumps, asymptotes), Corners, Cusps, and Vertical Tangents. Students will use one-sided limits of the difference quotient to prove non-differentiability analytically.
Purpose: To prevent students from blindly applying derivative rules to functions that aren't "well-behaved." It builds the logical foundation for future theorems (like Mean Value Theorem) which require both continuity and differentiability as prerequisites.
DOK Level 1 (Recall): Stating the theorem: "If f is differentiable at a, then f is continuous at a.
"DOK Level 2 (Skill/Concept): Identifying points on a graph where a function is continuous but not differentiable.
DOK Level 3 (Strategic Thinking): Solving for constants (a and b) in a piecewise function to make the function both continuous and differentiable at a specific point.
For Struggling Learners (Scaffolding):The "Zoom" Test: Use a graphing calculator to show that if you zoom in on a "smooth" curve, it eventually looks like a straight line (locally linear). If you zoom in on a corner (like |x|), it never looks like a single line.
For Advanced Learners (Extension):
Weierstrass Function: Introduce the concept of a function that is continuous everywhere but differentiable nowhere (visual/conceptual overview only).
College Board AP Classroom Assessments