Mathematics · 11th Grade · Sequences, Series, and Limits · Weeks 28-36

Continuity of Functions

Students will define continuity and identify different types of discontinuities in functions.

About This Topic

Continuity is one of the foundational ideas bridging precalculus and calculus. A function is continuous at a point if the function is defined there, the limit exists, and the limit equals the function value. When any one of these three conditions fails, a discontinuity occurs. Students in 11th grade encounter this concept as they begin working with limits, and understanding continuity helps them recognize where function behavior is predictable versus where it breaks down.

There are two main types of discontinuities students need to distinguish. Removable discontinuities, often called holes, occur where the limit exists but the function is either undefined or defined differently at that point. Non-removable discontinuities, such as jump discontinuities and infinite discontinuities (vertical asymptotes), cannot be fixed by simply redefining the function at a single point.

Active learning deepens this topic because students must do more than apply a checklist. Analyzing graphs, constructing piecewise functions with intentional discontinuities, and arguing about real-world models where breaks in continuity have consequences all push students toward genuine conceptual understanding.

Key Questions

1. Explain the three conditions required for a function to be continuous at a point.
2. Differentiate between removable and non-removable discontinuities.
3. Analyze the implications of discontinuity in real-world models.

Learning Objectives

• Explain the three conditions necessary for a function to be continuous at a specific point.
• Classify discontinuities as removable (holes) or non-removable (jumps, vertical asymptotes).
• Analyze graphs of functions to identify and describe points of discontinuity.
• Construct piecewise functions that exhibit specific types of discontinuities.
• Compare the behavior of continuous versus discontinuous functions in mathematical models.

Before You Start

Limits of Functions

Why: Students must understand the concept of a limit and how to calculate it to evaluate the second condition for continuity.

Function Evaluation

Why: Students need to be able to find the value of a function at a specific point to check the first and third conditions for continuity.

Graphing Functions

Why: Visualizing functions and their behavior is crucial for identifying and understanding different types of discontinuities.

Key Vocabulary

Continuity	A function is continuous at a point if it is defined at that point, its limit exists at that point, and the limit equals the function's value at that point.
Discontinuity	A point where a function fails to be continuous, meaning at least one of the three continuity conditions is not met.
Removable Discontinuity	A discontinuity that can be 'fixed' by redefining the function at a single point; often appears as a hole in the graph where the limit exists.
Non-removable Discontinuity	A discontinuity that cannot be fixed by redefining the function at a single point; includes jump discontinuities and infinite discontinuities (vertical asymptotes).
Limit	The value that a function or sequence approaches as the input or index approaches some value; the function does not necessarily have to equal this value.

Watch Out for These Misconceptions

Common Misconception

Students think a function defined at every point must be continuous everywhere.

What to Teach Instead

Show a piecewise function where f(a) is defined but the limit from the left and right do not agree (jump discontinuity). Having students construct such functions themselves during collaborative tasks makes this distinction concrete.

Common Misconception

Students confuse removable discontinuities with continuity, assuming that if a hole can be 'patched,' the function is already continuous.

What to Teach Instead

Use the three-condition checklist as a non-negotiable routine. Peer discussion during sorting activities helps students articulate why a function with a hole fails the third condition even if the first two pass.

Common Misconception

Students assume that if a limit exists at a point, the function must be continuous there.

What to Teach Instead

Present cases where the limit equals a value different from f(a), or where f(a) is undefined. Checking all three conditions as a group routine prevents students from stopping at the limit step.

Active Learning Ideas

Think-Pair-Share: Three-Condition Checklist

Present students with four function graphs, some continuous and some not. Students individually check each of the three continuity conditions for a specified point, then pair up to compare conclusions and resolve disagreements before class discussion.

25 min·Pairs

Sorting Activity: Classify the Discontinuity

Small groups receive cards with function graphs and algebraic expressions showing various discontinuities. They sort the cards into three categories: removable, jump, and infinite discontinuity, then write one sentence explaining the distinguishing feature of each category.

30 min·Small Groups

Gallery Walk: Real-World Discontinuity Models

Post four scenarios around the room (water flow stopping suddenly, tax brackets, postage rates, phone signal loss). Groups rotate and sketch a graph for each context, then label the type of discontinuity and explain what the break means in that context.

35 min·Small Groups

Build-a-Break: Constructing Piecewise Functions

Pairs design a piecewise function that contains one removable and one non-removable discontinuity, then trade with another pair to verify the discontinuities algebraically and graphically. Each group presents their function and explains their design choices.

40 min·Pairs

Real-World Connections

• Engineers designing bridges or aircraft wings must ensure their material stress models are continuous over the expected load ranges to avoid catastrophic failure.
• Economists analyzing stock market data look for points of discontinuity that might indicate market crashes or significant policy shifts, which can cause sudden jumps or drops in value.
• Medical professionals interpreting patient vital signs, such as heart rate or blood pressure, expect continuity under normal conditions; significant jumps or drops signal potential health issues.

Assessment Ideas

Exit Ticket

Provide students with three function definitions: f(x) = x^2, g(x) = 1/x, and h(x) = {x if x != 0, 1 if x = 0}. Ask them to identify which function is continuous at x=0 and explain why, referencing the three conditions. For the discontinuous functions, ask them to name the type of discontinuity.

Quick Check

Display graphs of functions with various discontinuities. Ask students to write down the x-values where discontinuities occur and classify each as removable or non-removable. Review answers as a class, asking students to justify their classifications.

Discussion Prompt

Pose the question: 'Imagine a function describing the temperature in a room over a 24-hour period. Can this function have any discontinuities? If so, where and why?' Guide students to discuss scenarios like a thermostat turning on/off or a sudden opening of a door.

Frequently Asked Questions

What are the three conditions for a function to be continuous at a point?

A function f is continuous at x = a if: (1) f(a) is defined, (2) the limit of f(x) as x approaches a exists, and (3) the limit equals f(a). All three must hold. If any one fails, the function has a discontinuity at that point.

What is the difference between a removable and non-removable discontinuity?

A removable discontinuity is a hole in the graph where the limit exists but the function value is missing or different. It can be fixed by redefining f at that one point. Non-removable discontinuities, like jumps or vertical asymptotes, cannot be repaired this way because the limit either does not exist or is infinite.

Where does continuity appear in real-world situations?

Tax brackets create jump discontinuities at threshold income levels. Postage rate tables jump at weight boundaries. Physical quantities like temperature or population are usually modeled as continuous because they change smoothly. Recognizing where real models break continuity helps students understand why the concept matters outside of math class.

How does active learning support understanding of continuity in precalculus?

Continuity involves three simultaneous conditions, and students frequently forget to check all of them. Sorting activities and peer-checking routines build the habit of applying each condition systematically. When students construct piecewise functions with intentional breaks and explain them to peers, they internalize the concept at a level that lecture alone rarely achieves.