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Three Conditions for Continuity Visualized

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Description

This MicroSim visualizes the three conditions required for a function to be continuous at a point c:

  1. f(c) is defined - The function has a value at the point
  2. The limit exists - Both left and right limits exist and are equal
  3. The limit equals f(c) - The function value matches the limit

Watch Delta, our triangular calculus robot mascot, as she tries to navigate along each function. When she can smoothly travel through a point, the function is continuous there. When she encounters a gap, jump, or misplaced landing spot, the function is discontinuous.

Five Scenarios

Use the A-E buttons to explore different discontinuity types:

  • A: Continuous - All three conditions pass. Delta travels smoothly.
  • B: Hole - f(c) is undefined. Delta hovers over empty space.
  • C: Jump - The left and right limits differ. Delta would have to teleport.
  • D: Misplaced Point - f(c) exists but doesn't match the limit. The landing spot is in the wrong place.
  • E: Vertical Asymptote - The limit doesn't exist at all. Delta sees infinity.

How to Use

  1. Select a scenario (A-E) to see different discontinuity types
  2. Use the slider to move the target point c along the x-axis
  3. Click "Check at c" to watch the three conditions be evaluated one by one
  4. See the final verdict: CONTINUOUS (green) or DISCONTINUOUS (red)
  5. Click "Reset" to try again

Delta Moment

"Can I walk here smoothly, or is there a gap I'd fall through? That's what continuity is all about! I need three things: somewhere to land, a clear path from both sides, and that path has to actually reach my landing spot."

Lesson Plan

Learning Objective

Students will explain how each of the three continuity conditions corresponds to visual features of a function graph.

Grade Level

High School (AP Calculus) or Early Undergraduate

Prerequisites

  • Understanding of function notation f(x)
  • Basic concept of limits (left-hand, right-hand)
  • Familiarity with coordinate graphing

Duration

15-20 minutes

Activities

Warm-up (3 minutes): Ask students: "What does it mean intuitively for a graph to be 'continuous'?" Gather responses like "no breaks," "can draw without lifting pencil," etc.

Exploration (10 minutes):

  1. Start with Scenario A (continuous). Have students predict what will happen before clicking "Check at c." Verify all three conditions pass.

  2. Move to Scenario B (hole). Ask: "Which condition do you think will fail?" Check and discuss why f(c) being undefined breaks continuity.

  3. Explore Scenarios C, D, and E. For each:

  4. Predict which condition(s) will fail
  5. Verify with the MicroSim
  6. Discuss the visual feature that causes the failure

Discussion (5 minutes):

  • "Why do we need all THREE conditions? Why isn't just having f(c) defined enough?"
  • "Which type of discontinuity (hole, jump, asymptote) seems most 'severe'? Why?"
  • "Can you think of real-world situations where each type might occur?"

Assessment

Have students draw their own function with a discontinuity at x = 3 and identify which of the three conditions fails. Trade drawings with a partner and verify each other's analysis.

Extensions

  • Challenge: Create a function that fails ONLY condition 3 (like Scenario D)
  • Research: Find real-world functions that have discontinuities (step functions in tax brackets, voltage switches, etc.)

References

  1. Wikipedia: Continuous Function - Comprehensive overview of continuity in mathematics
  2. Khan Academy: Continuity at a Point - Video explanation with examples
  3. Paul's Online Math Notes: Continuity - Detailed notes with worked examples