Continuity of Functions

Active learning works well for continuity because students need to physically examine where functions break down rather than passively watch graphs or listen to explanations. This topic requires spatial reasoning and careful reading of function definitions, both of which improve when students manipulate examples with their own hands and discuss them in small groups.

• 11th Grade
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

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

Think-Pair-Share

⏱ 25 min·Pairs

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.

Prepare & details

Explain the three conditions required for a function to be continuous at a point.

Facilitation Tip: During the Think-Pair-Share, provide each pair with a laminated three-condition checklist to hold up when presenting, so the routine becomes visual and tangible.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Document Mystery

⏱ 30 min·Small Groups

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.

Prepare & details

Differentiate between removable and non-removable discontinuities.

Facilitation Tip: For the Sorting Activity, assign roles within groups (reader, recorder, presenter) to ensure every student engages with the examples rather than one student dominating.

Setup: Groups at tables with document sets

Materials: Document packet (5-8 sources), Analysis worksheet, Theory-building template

• Analyze
• Evaluate
• Self-Management
• Decision-Making

Gallery Walk

⏱ 35 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.

Prepare & details

Analyze the implications of discontinuity in real-world models.

Facilitation Tip: In the Gallery Walk, place the real-world models at eye level and ask students to annotate each station with sticky notes that name the type of discontinuity and explain why it matters in context.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Document Mystery

⏱ 40 min·Pairs

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.

Prepare & details

Explain the three conditions required for a function to be continuous at a point.

Facilitation Tip: When students Build-a-Break, require them to write a brief justification on the back of each piecewise segment that explains which condition fails at the break point.

Setup: Groups at tables with document sets

Materials: Document packet (5-8 sources), Analysis worksheet, Theory-building template

• Analyze
• Evaluate
• Self-Management
• Decision-Making

Teaching This Topic

Teach continuity by making the three-condition checklist non-negotiable. Start every activity with it, refer to it in explanations, and require students to reference it in their work. Avoid teaching continuity as a single definition; instead, emphasize that it is a set of conditions that must all be true simultaneously. Research shows that students grasp discontinuities more easily when they see examples where one condition fails at a time, so design tasks that isolate each condition before combining them.

What to Expect

Successful learning looks like students accurately identifying the three conditions for continuity at a point, distinguishing between types of discontinuities, and explaining their reasoning using precise mathematical language. You will see this in their justifications during discussions and in their written work that references the checklist and examples.

Watch Out for These Misconceptions

Common Misconception

During the Build-a-Break activity, watch for students who create piecewise functions where f(a) is defined but the left and right limits do not match. They may assume the function is continuous because it is defined everywhere.

What to Teach Instead

Have students present their piecewise functions to the class and ask the class to verify the three conditions using the checklist. Peer questioning will reveal that the limit does not exist at the break point, so the function fails the second condition.

Common Misconception

During the Sorting Activity: Classify the Discontinuity, students may call a removable discontinuity 'continuous' because the hole can be patched.

What to Teach Instead

Ask students to write the function value f(a) on the sorting card and compare it to the limit value. During the group discussion, have students explain why a hole means the third condition fails, even if the first two conditions are met.

Common Misconception

During the Think-Pair-Share: Three-Condition Checklist, students may stop once they verify that the limit exists, assuming continuity is guaranteed.

What to Teach Instead

Require students to explicitly state all three conditions in their pair share, even if the first two are obviously true. Circulate and prompt students with, 'You’ve found the limit. Does that mean the function is continuous here? What else do you need to check?'

Assessment Ideas

Exit Ticket

After the Think-Pair-Share: Three-Condition Checklist, collect students’ written explanations for why each function in the exit ticket is continuous or discontinuous at x=0. Focus on whether they correctly name the violated condition and the type of discontinuity.

Quick Check

During the Sorting Activity: Classify the Discontinuity, collect students’ classification sheets and check for accurate pairing of function cards with discontinuity types. Use this to identify patterns in misclassifications for targeted review.

Discussion Prompt

After the Gallery Walk: Real-World Discontinuity Models, ask students to share one real-world example they observed and explain which type of discontinuity it represents. Listen for students to connect the type of discontinuity to the three conditions and the context of the example.

Extensions & Scaffolding

• Challenge early finishers to create a function that has a removable discontinuity at x=2 and a non-removable discontinuity at x=5.
• Scaffolding for struggling students: Provide partially completed piecewise functions where students only need to adjust one segment to create a discontinuity, then ask them to explain which condition is violated.
• Deeper exploration: Ask students to research and present on a real-world phenomenon (e.g., stock market fluctuations, heartbeat monitoring) that exhibits a discontinuity, and classify the type of discontinuity mathematically.

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.