Unit 03 · Lesson 03 · Week 7 · Continuity & Asymptotic Behavior
Piecewise Functions and Continuity Conditions
Continuity at an input requires an existing function value, an existing limit, and agreement between them. Continuity on an interval supports global conclusions such as the Intermediate Value Theorem. This lesson emphasizes piecewise functions and continuity conditions through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in piecewise functions and continuity conditions.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do continuity conditions connect local behavior, global guarantees, and model reliability?
Visual model
Continuity at an input requires an existing function value, an existing limit, and agreement between them. Continuity on an interval supports global conclusions such as the Intermediate Value Theorem.
\lim_{x\to a}f(x)=f(a)f(c)=Na<c<b
Worked reasoning
Misconception check: The Intermediate Value Theorem guarantees that a value occurs; it does not identify where it occurs or guarantee uniqueness.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Piecewise Functions and Continuity Conditions
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
What value of c makes f(x)=(x²−4)/(x−2) for x≠2, and f(2)=c continuous?
Answer: 4
Reasoning: The limit is x+2 at x=2, giving 4.
Reasoning: The limit is x+2 at x=2, giving 4.
List the three conditions for continuity at x=a.
Answer: f(a) exists; lim x→a f(x) exists; the limit equals f(a).
Reasoning: All three are required.
Reasoning: All three are required.
Classify a finite hole in a graph.
Answer: Removable discontinuity
Reasoning: Defining the missing point appropriately can restore continuity.
Reasoning: Defining the missing point appropriately can restore continuity.
Does a jump discontinuity have a two-sided limit?
Answer: No
Reasoning: The one-sided limits differ.
Reasoning: The one-sided limits differ.
What does the Intermediate Value Theorem guarantee?
Answer: Every intermediate output occurs on a continuous interval.
Reasoning: Continuity connects endpoint outputs without gaps.
Reasoning: Continuity connects endpoint outputs without gaps.
Why must continuity be checked before using IVT?
Answer: The theorem's guarantee depends on continuity.
Reasoning: A discontinuity can skip intermediate values.
Reasoning: A discontinuity can skip intermediate values.