Differentiability and its Relation to Continuity

Active learning helps students grasp differentiability and continuity because these concepts are best understood through visual and kinesthetic experiences. When students plot graphs, debate tangent slopes, and construct counterexamples, they move beyond abstract definitions to see how sharp corners, smooth curves, and jumps behave mathematically.

• Class 12
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Explain why differentiability at a point implies continuity at that point, using the formal definition of a derivative.
2. Compare and contrast functions that are continuous but not differentiable at a point with those that are differentiable, identifying key graphical features.
3. Analyze the geometric meaning of the derivative as the slope of the tangent line to a curve at a given point.
4. Construct a piecewise function that is continuous at a specific point but not differentiable there, justifying the construction with limit calculations.
5. Calculate the left-hand and right-hand derivatives for a given function at a point to determine differentiability.

Problem-Based Learning

⏱ 30 min·Pairs

Pair Graphing: Test Continuity and Differentiability

Pairs graph f(x) = |x| and f(x) = x sin(1/x) for x ≠ 0, f(0)=0 on graph paper. They mark points, compute left and right limits visually, and note where tangents fail. Discuss findings with the class.

Prepare & details

Explain why differentiability implies continuity, but continuity does not imply differentiability.

Facilitation Tip: During Pair Graphing, circulate and ask each pair to explain why their chosen function is not differentiable at a specific point, ensuring they use the tangent slope test rather than just pointing to the graph.

Setup: Standard classroom with movable furniture arranged for groups of 5 to 6; if furniture is fixed, groups work within rows using a designated recorder. A blackboard or whiteboard for capturing the whole-class 'need-to-know' list is essential.

Materials: Printed problem scenario cards (one per group), Structured analysis templates: 'What we know / What we need to find out / Our hypothesis', Role cards (recorder, researcher, presenter, timekeeper), Access to NCERT textbooks and any supplementary reference materials, Individual reflection sheets or exit slips with a board-exam-style application question

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 45 min·Small Groups

Small Groups: Construct Counterexamples

Groups create functions continuous but not differentiable at x=0, such as piecewise or absolute value variants. They plot, verify continuity via limits, and show derivative limit fails. Present one example per group.

Prepare & details

Analyze the geometric interpretation of a derivative as the slope of a tangent line.

Facilitation Tip: While constructing counterexamples in small groups, remind students to justify their functions using difference quotient limits, not just visual features.

Setup: Standard classroom with movable furniture arranged for groups of 5 to 6; if furniture is fixed, groups work within rows using a designated recorder. A blackboard or whiteboard for capturing the whole-class 'need-to-know' list is essential.

Materials: Printed problem scenario cards (one per group), Structured analysis templates: 'What we know / What we need to find out / Our hypothesis', Role cards (recorder, researcher, presenter, timekeeper), Access to NCERT textbooks and any supplementary reference materials, Individual reflection sheets or exit slips with a board-exam-style application question

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 35 min·Whole Class

Whole Class: Tangent Slope Debate

Project graphs of continuous non-differentiable functions. Class votes on tangent existence at key points, then computes difference quotients step-by-step on board. Tally and resolve disagreements.

Prepare & details

Construct a function that is continuous but not differentiable at a specific point.

Facilitation Tip: In the Tangent Slope Debate, assign roles like ‘defender of continuity’ or ‘advocate of differentiability’ to push students to articulate nuanced arguments using graphs and calculations.

Setup: Standard classroom with movable furniture arranged for groups of 5 to 6; if furniture is fixed, groups work within rows using a designated recorder. A blackboard or whiteboard for capturing the whole-class 'need-to-know' list is essential.

Materials: Printed problem scenario cards (one per group), Structured analysis templates: 'What we know / What we need to find out / Our hypothesis', Role cards (recorder, researcher, presenter, timekeeper), Access to NCERT textbooks and any supplementary reference materials, Individual reflection sheets or exit slips with a board-exam-style application question

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 25 min·Individual

Individual: Limit Worksheet

Students compute left/right derivatives for given functions at specified points using tables. Identify differentiable points and justify with epsilon-delta hints. Share one solution in plenary.

Prepare & details

Explain why differentiability implies continuity, but continuity does not imply differentiability.

Facilitation Tip: For the Limit Worksheet, encourage students to write both the left-hand and right-hand limits explicitly before concluding differentiability.

Setup: Standard classroom with movable furniture arranged for groups of 5 to 6; if furniture is fixed, groups work within rows using a designated recorder. A blackboard or whiteboard for capturing the whole-class 'need-to-know' list is essential.

Materials: Printed problem scenario cards (one per group), Structured analysis templates: 'What we know / What we need to find out / Our hypothesis', Role cards (recorder, researcher, presenter, timekeeper), Access to NCERT textbooks and any supplementary reference materials, Individual reflection sheets or exit slips with a board-exam-style application question

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Teaching This Topic

Start by anchoring the concept in familiar motions: a smooth ride implies continuity and differentiability, while a sharp turn or a sudden stop breaks one or both. Avoid rushing into formal proofs; instead, use graphs and real-world examples to build intuition first. Research shows that students retain these ideas better when they physically plot points and calculate slopes, so prioritize hands-on work over lectures. Watch for students who confuse the existence of a tangent line with the continuity of the derivative function itself.

What to Expect

Students will confidently distinguish between continuity and differentiability by identifying tangent slopes, sharp corners, and removable discontinuities on graphs. They should explain why differentiable functions are continuous but not all continuous functions are differentiable, using precise language and examples from their activities.

Watch Out for These Misconceptions

Common Misconception

During Pair Graphing, watch for students who assume continuity implies differentiability because the graph ‘looks smooth’ at most points.

What to Teach Instead

Ask pairs to calculate the difference quotient limits at the corner point of their chosen function, specifically checking if the left and right limits match, to redirect their reasoning through calculation.

Common Misconception

During Construct Counterexamples, watch for students who believe a continuous derivative is required for differentiability at a point.

What to Teach Instead

Have groups test functions like f(x) = x^2 * sin(1/x) for x ≠ 0 and f(0) = 0, showing how the derivative exists at x=0 but is not continuous there.

Common Misconception

During Tangent Slope Debate, watch for students who conflate continuous functions with differentiable ones based on equal one-sided limits.

What to Teach Instead

Use the debate to introduce the formal definition of differentiability, asking students to compute the difference quotient limits and compare them to the function’s continuity at the point.

Assessment Ideas

Exit Ticket

After Pair Graphing, provide three unlabeled function graphs and ask students to label each as differentiable, continuous but not differentiable, or discontinuous, with a one-sentence justification referencing tangent slopes or sharp corners.

Quick Check

After Construct Counterexamples, give students the function f(x) = |x - 1| + 2 and ask them to compute the left-hand and right-hand derivatives at x = 1, then state whether the function is differentiable there and explain their reasoning using the difference quotient.

Discussion Prompt

During Tangent Slope Debate, pose the question: 'If a car’s position is continuous, does its velocity have to be continuous?' Have students discuss how non-differentiable points (like sharp turns) affect motion and what this means for real-world applications like smooth rides.

Extensions & Scaffolding

• Challenge students to find a function that is continuous everywhere but differentiable nowhere, and present its graph with explanations to the class.
• For students struggling with sharp corners, provide pre-drawn graphs of functions like x^(1/3) or x*|x| and ask them to compute one-sided derivatives at the critical point.
• Allow extra time for students to explore functions like f(x) = x*sin(1/x) near x=0, discussing continuity and differentiability with teacher guidance.

Key Vocabulary

Differentiability	A function is differentiable at a point if its derivative exists at that point, meaning the function has a unique, non-vertical tangent line.
Continuity	A function is continuous at a point if its graph can be drawn through that point without lifting the pen, meaning the limit exists, the function value exists, and they are equal.
Derivative	The instantaneous rate of change of a function with respect to its variable, geometrically represented as the slope of the tangent line.
Tangent Line	A straight line that touches a curve at a single point without crossing it at that point, representing the instantaneous direction of the curve.
Cusp	A point on a curve where the tangent line is vertical, or where the curve has a sharp point, indicating continuity but not differentiability.