Activity 01
Pair Graphing: Spot the Discontinuity
Pairs receive printed graphs of functions like step or rational types. They mark left and right approaches to points, note limit values, and classify continuity. Discuss findings with the class, justifying with epsilon-delta hints if ready.
Analyze the role of one-sided limits in defining the continuity of a function.
Facilitation TipDuring Pair Graphing, have students physically trace the graph with their fingers from left and right to reinforce the idea of approaching a point from both sides.
What to look forPresent students with the graph of a piecewise function. Ask them to: 1. Write down the left-hand limit as x approaches the point of interest. 2. Write down the right-hand limit as x approaches the point of interest. 3. State whether the function is continuous at that point and justify their answer.
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Activity 02
Small Groups: Function Factory
Groups create three functions: one continuous, one with jump discontinuity, one removable. Use piecewise definitions on paper or Desmos. Present to class, peers vote on continuity and explain one-sided limits.
Compare and contrast continuous and discontinuous functions using graphical examples.
Facilitation TipIn Function Factory, ask groups to trade their piecewise functions with another group and verify each other’s limits to encourage peer accountability.
What to look forProvide each student with a function definition, e.g., $f(x) = \begin{cases} x² & if x < 2 \\ 5 & if x = 2 \\ 2x & if x > 2 \end{cases}. Ask them to calculate f(2), lim f(x), and lim f(x). Then, ask them to conclude if f(x) is continuous at x=2$ and explain why or why not.
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Activity 03
Whole Class: Limit Chase
Project a function graph. Class calls out left/right limits at points as you zoom. Vote on continuity, then reveal table of values to confirm. Repeat with student-submitted graphs.
Justify why a function might be discontinuous at a specific point.
Facilitation TipFor Limit Chase, stand at the back of the room and quietly observe which students hesitate at jump discontinuities; those are the ones who need targeted questioning during the debrief.
What to look forPose the question: 'Imagine a function representing the temperature in a room over a 24-hour period. Can this function have a jump discontinuity? If so, where and why might it occur? If not, why must it be continuous?' Facilitate a class discussion where students use their understanding of one-sided limits and continuity to justify their answers.
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Activity 04
Individual: Continuity Checklist
Students get worksheets with 10 functions. For each, compute one-sided limits, check f(a), and tick continuous or not. Share one tricky case in pairs for verification.
Analyze the role of one-sided limits in defining the continuity of a function.
Facilitation TipWhen students complete the Continuity Checklist, collect a few responses at random and read them aloud to the class for immediate feedback on precision.
What to look forPresent students with the graph of a piecewise function. Ask them to: 1. Write down the left-hand limit as x approaches the point of interest. 2. Write down the right-hand limit as x approaches the point of interest. 3. State whether the function is continuous at that point and justify their answer.
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Generate Complete Lesson→A few notes on teaching this unit
Experienced teachers approach this topic by building from concrete to abstract: start with graphs where students spot discontinuities visually, then connect those visuals to limit notation, and finally formalise the continuity condition. Avoid rushing into epsilon-delta definitions; instead, use multiple examples where students adjust function values at a point to ‘fix’ discontinuities. Research from calculus reform shows that students grasp limits better when they first experience them through motion—tracing graphs with hands or imagining a point moving along the curve—before writing symbols.
By the end of these activities, students should confidently explain why both left and right limits matter for continuity. They should also be able to identify discontinuities on graphs and justify their answers using limit values and function values. Success looks like students correcting each other’s reasoning during discussions and using precise language like 'limit exists' and 'matches f(a)' without prompting.
Watch Out for These Misconceptions
During Pair Graphing, watch for students who assume continuity if the graph looks smooth from one side only.
Remind pairs to always trace both sides of the point and record the left and right limits separately on their sheets before deciding continuity.
During Function Factory, watch for students who ignore the function value at the point when checking continuity.
Prompt groups to plot f(a) on the same number line as the left and right limits and ask, 'Does this point sit exactly where the limits meet?'
During Limit Chase, watch for students who think any break in the graph means the function is not differentiable anywhere.
Pause the game at a jump discontinuity and ask, 'If you were to draw a tangent line at this point, where would it go?' to highlight that only the point of discontinuity is affected.
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