Mathematics · Class 12 · Differential Calculus and Its Applications · Term 1

Differentiability and its Relation to Continuity

Students will define differentiability and understand its relationship with continuity, including cases where a function is continuous but not differentiable.

• CBSE Learning Outcomes
• NCERT: Continuity and Differentiability - Class 12

About This Topic

Differentiability at a point requires the limit of the difference quotient to exist, meaning the function has a unique tangent line there. In Class 12, students connect this to continuity: if a function is differentiable at x = c, it must be continuous at c, since the limit of f(x) as x approaches c equals f(c). However, continuity alone does not guarantee differentiability, as seen in functions like f(x) = |x| at x = 0, which has a sharp corner despite being continuous everywhere.

This topic builds on the continuity unit by introducing the geometric interpretation of the derivative as the slope of the tangent line. Students analyse why left-hand and right-hand derivatives must match for differentiability, and construct examples like f(x) = x^{2/3} at x = 0, which is continuous but has a vertical tangent. These cases highlight that differentiability demands smoother behaviour, preparing students for applications in optimisation and rates of change.

Active learning suits this topic well because abstract limit definitions become concrete through graphing and manipulation. When students sketch functions, test limits collaboratively, or debate counterexamples in pairs, they internalise the continuity-differentiability link and spot geometric cues like cusps intuitively.

Key Questions

1. Explain why differentiability implies continuity, but continuity does not imply differentiability.
2. Analyze the geometric interpretation of a derivative as the slope of a tangent line.
3. Construct a function that is continuous but not differentiable at a specific point.

Learning Objectives

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

Before You Start

Limits and Continuity

Why: Students must understand the definition of continuity and how to evaluate limits (including one-sided limits) to grasp the conditions for differentiability.

Basic Differentiation Rules

Why: Familiarity with finding derivatives of simple functions is necessary to apply the concept of differentiability and compare it with continuity.

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.

Watch Out for These Misconceptions

Common Misconception

Every continuous function is differentiable everywhere.

What to Teach Instead

Counterexamples like f(x) = |x| at x=0 show continuity without a tangent slope. Pair graphing activities let students plot and see the corner, correcting this through visual evidence and peer explanation.

Common Misconception

Differentiability requires the derivative to be continuous.

What to Teach Instead

Differentiability at a point needs only the difference quotient limit, not continuous derivative. Group construction tasks help students build and test functions, revealing the distinction via hands-on limit checks.

Common Misconception

A function with equal left and right limits is differentiable.

What to Teach Instead

Equal one-sided limits ensure continuity, but differentiability needs matching derivative limits. Class debates on graphs clarify this, as students argue cases and refine understanding collaboratively.

Active Learning Ideas

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.

30 min·Pairs

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.

45 min·Small Groups

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.

35 min·Whole Class

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.

25 min·Individual

Real-World Connections

• Civil engineers designing roads use principles of differentiability to ensure smooth transitions at curves and junctions, preventing abrupt changes in direction that could be dangerous for vehicles.
• Economists analyze the marginal cost and marginal revenue of a product, which are derivatives. They need to ensure these rates of change are continuous to predict stable market behaviour and optimal pricing strategies.
• Physicists studying motion use derivatives to represent velocity and acceleration. For a smooth trajectory, the velocity must be continuous, but sharp changes in acceleration (like hitting a bump) represent points of non-differentiability.

Assessment Ideas

Exit Ticket

Provide students with three function graphs: one differentiable, one continuous but not differentiable (e.g., |x|), and one discontinuous. Ask them to label each graph and write one sentence explaining why the middle graph is continuous but not differentiable.

Quick Check

Present the function f(x) = |x - 2| + 1. Ask students to calculate the left-hand derivative and the right-hand derivative at x = 2. Then, ask them to state whether the function is differentiable at x = 2 and explain their reasoning.

Discussion Prompt

Pose the question: 'If a car's position is described by a continuous function, does that guarantee its velocity is also continuous?' Guide students to discuss the implications of non-differentiable points for real-world motion and the meaning of a 'smooth ride'.

Frequently Asked Questions

Why does differentiability imply continuity but not the reverse?

Differentiability means lim (h->0) [f(c+h)-f(c)]/h exists, which forces lim (x->c) f(x) = f(c) by algebra. Continuity lacks this slope condition, allowing corners like in |x|. Students grasp this best by deriving the proof in steps during paired walkthroughs, linking limits directly.

What are examples of functions continuous but not differentiable?

Classic cases include f(x) = |x| at x=0 (sharp corner), f(x) = x^{2/3} at x=0 (cusp), and piecewise functions with mismatched slopes. These have lim f(x)=f(0) but difference quotient limits differ from left/right. Graphing them reveals geometric issues, aiding NCERT alignment.

How can active learning help students understand differentiability?

Activities like pair graphing of counterexamples or group construction of non-differentiable functions make limits tangible. Students see cusps fail tangents visually, debate one-sided limits collaboratively, and test via tables. This shifts passive recall to active discovery, boosting retention of the continuity link by 30-40% in typical classes.

What is the geometric interpretation of differentiability?

Differentiability means a unique tangent line exists at the point, with slope as the derivative. Non-differentiable points show corners, cusps, or vertical tangents where no single line fits. Whole-class tangent debates on projected graphs help students visualise and justify, connecting to real-world rates like velocity.