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Piecewise Continuity Explorer

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Description

This MicroSim helps students determine whether piecewise functions are continuous at their boundary points by systematically checking all three conditions for continuity:

  1. The function is defined at the boundary point: f(a) exists
  2. The limit exists at the boundary point: The left-hand limit equals the right-hand limit
  3. The function value equals the limit: f(a) = lim(x->a) f(x)

Visual Elements

  • Two-colored graph: The left piece of the function is shown in blue, the right piece in orange
  • Approach indicators: Colored dots show values approaching the boundary from each side
  • Boundary analysis: Green dot shows the actual function value at the boundary point
  • Info panel: Displays the left limit, right limit, and function value
  • Condition checklist: Animates through verification of each continuity condition
  • Verdict display: Shows "CONTINUOUS" or "DISCONTINUOUS" based on the analysis

Interactive Controls

  • Evaluate Continuity: Animates the approach to the boundary and steps through the three conditions
  • Reset: Returns to the initial state
  • Select Example: Dropdown menu to choose from four preset examples:
    • Continuous Piecewise (all conditions met)
    • Jump Discontinuity (limits do not match)
    • Removable Discontinuity (function value does not equal limit)
    • Challenge Case (a non-obvious continuous case)
  • Approach slider: Manually adjust how close the approach points are to the boundary

Delta Moment

"Can I walk here smoothly, or is there a gap I'd fall through? That's what continuity is all about. If the left side and right side meet up perfectly AND the function actually touches that meeting point... I can roll right through!"

Lesson Plan

Learning Objective

Students will be able to determine whether piecewise functions are continuous at boundary points by checking all three conditions for continuity.

Warm-Up (5 minutes)

  1. Ask students: "What does it mean for a function to be continuous?"
  2. Have students sketch a function that is NOT continuous and explain why

Guided Exploration (15 minutes)

  1. Open the MicroSim and start with Example 1 (Continuous Piecewise)
  2. Click "Evaluate Continuity" and observe each condition being checked
  3. Discuss:
  4. What does the blue dot represent? (Left-hand limit approach)
  5. What does the orange dot represent? (Right-hand limit approach)

  6. What does the green dot represent? (Actual function value)

  7. Switch to Example 2 (Jump Discontinuity)

  8. Before clicking Evaluate, ask students to predict: Will this be continuous?
  9. Run the evaluation and discuss which condition fails

Independent Practice (10 minutes)

Have students work through Examples 3 and 4:

  • For each example, FIRST predict the outcome
  • THEN verify using the MicroSim
  • Write down which condition(s) fail for discontinuous cases

Challenge Questions

  1. Can a function be discontinuous if the limit exists at a point?
  2. If both one-sided limits equal each other, is the function automatically continuous?
  3. Create your own piecewise function that has a removable discontinuity at x = 1

Assessment

Students should be able to:

  • [ ] State all three conditions for continuity
  • [ ] Identify which condition fails for a given discontinuous function
  • [ ] Distinguish between jump and removable discontinuities
  • [ ] Explain why all three conditions are necessary

References