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CBSE Class 12 Mathematics Chapter 5 Continuity and Differentiability Worksheet with Answers

Continuity and Differentiability is Chapter 5 in NCERT Class 12 Mathematics and typically carries 8-10 marks in the CBSE board examination. Mastery of this chapter requires both conceptual clarity and rigorous practice. This worksheet has been carefully designed to cover all major topics—continuity at a point, differentiability, chain rule, derivatives of composite and inverse trigonometric functions—and is structured to mirror the 2025 CBSE question-paper format. Print it out, set a timer for 90 minutes, and solve it under exam conditions for maximum benefit.

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Key takeaways

✓This worksheet covers all essential NCERT Class 12 Mathematics Chapter 5 topics including continuity, differentiability, chain rule, and derivatives of inverse functions.
✓Difficulty level is mixed—20% easy, 50% moderate, 30% challenging—matching the CBSE Class 12 board exam pattern for 2025.
✓The worksheet contains 25+ questions across MCQs, fill-in-the-blanks, short-answer, long-answer, and one case-study problem.
✓Complete answer key with step-by-step explanations is provided at the end for effective self-assessment and learning.
✓Suggested time limit is 90 minutes; attempting it under timed conditions simulates actual board exam pressure.
✓Regular practice with such worksheets improves accuracy in applying the chain rule and identifying points of discontinuity.
✓Parents can download and print this worksheet for offline practice, ideal for revision during board exam preparation.

Quick Chapter Recap: Continuity and Differentiability

Before you begin the worksheet, recall that a function f(x) is continuous at x = a if the limit of f(x) as x approaches a equals f(a). Formally, lim(x→a) f(x) = f(a). A function is differentiable at a point if the derivative exists at that point, which requires the left-hand and right-hand derivatives to be equal. Differentiability implies continuity, but the converse is not true—a function can be continuous yet not differentiable, as seen with f(x) = |x| at x = 0. The chain rule is central to this chapter: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx). You will also use derivatives of inverse trigonometric functions like d/dx(sin⁻¹x) = 1/√(1 – x²) and d/dx(tan⁻¹x) = 1/(1 + x²). Keep these formulae at your fingertips while solving.

Continuity: lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a)
Differentiability at x = a requires equality of left-hand derivative (LHD) and right-hand derivative (RHD)
Chain rule: dy/dx = (dy/du) × (du/dx) for composite functions
Every differentiable function is continuous, but not every continuous function is differentiable
Standard derivatives: d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(xⁿ) = n xⁿ⁻¹
Inverse trigonometric derivatives are non-negotiable for CBSE Class 12 board exams

Worksheet Instructions and Difficulty Level

This worksheet is designed for a 90-minute session and is split into five graded sections plus one case-study question. Attempt all questions in order: start with Section A (MCQs) to build confidence, then move through fill-in-the-blanks, matching, short answers, and finally the long HOTS questions. The difficulty distribution is 20% easy, 50% moderate, and 30% challenging, closely mirroring the 2025 CBSE Class 12 Mathematics board paper. Use a pencil for rough work and write final answers in pen. Do not refer to NCERT Class 12 Mathematics textbook or Class 12 Mathematics solutions during the timed attempt—treat this as a mini board exam. After 90 minutes, stop and use the answer key at the end to mark your work and identify weak areas. This self-assessment loop is critical for mastering Continuity and Differentiability before the boards.

Difficulty: Mixed (Easy 20%, Moderate 50%, Challenging 30%)
Suggested time: 90 minutes under exam conditions
Total questions: 25+ across MCQs, fill-in-the-blanks, short-answer, long-answer, and case-study
Marks distribution: Section A (6 marks), Section B (5 marks), Section C (4 marks), Section D (15 marks), Section E (15 marks), Case Study (5 marks)
Use only NCERT-approved formulae and standard notation
Keep a calculator handy only if permitted; most CBSE 12 Mathematics problems require algebraic simplification, not heavy computation

Section A: Multiple Choice Questions (1 mark each)

Each MCQ has four options; choose the single best answer. These questions test your conceptual understanding of continuity, differentiability, and the chain rule. Read each question twice and eliminate obviously wrong options before selecting your final answer. Remember that in CBSE Class 12 Mathematics Chapter 5, a common trap is confusing 'continuous but not differentiable' with 'differentiable but not continuous'—the latter is impossible. Work neatly in the margin and circle your final choice clearly.

Section B: Fill in the Blanks (1 mark each)

Write the exact numerical value or algebraic expression in the blank. Do not include extra words or units unless the question specifies. For Class 12 Mathematics Chapter 5, pay close attention to domain restrictions—for instance, sin⁻¹x is defined only for –1 ≤ x ≤ 1, and log x requires x > 0. If a function involves a parameter k or a, solve for it by applying the continuity or differentiability condition. These questions often appear in the CBSE board exam and carry easy marks if you know your formulae well.

Section C: Match the Following / True or False (1 mark each)

This section tests your ability to classify statements or pair concepts correctly. For match-the-following, write the correct pairing (e.g. A–III, B–I) in your answer sheet. For true/false, write T or F clearly—no half-marks for unclear handwriting in CBSE 12 Mathematics board exams. If a statement says 'Every continuous function is differentiable,' mark it False and recall the counter-example f(x) = |x|. Always justify your answer mentally even if the worksheet does not ask for a written justification; this habit solidifies conceptual clarity.

Section D: Short Answer Questions (3 marks each)

Each short-answer question carries 3 marks and expects you to show clear working over three to five lines. In the CBSE Class 12 Mathematics board exam, examiners award step-wise marks, so write every algebraic manipulation. Typical questions ask you to check continuity or differentiability at a given point, find dy/dx for a composite or inverse trigonometric function using the chain rule, or determine the value of a parameter that ensures continuity. State the condition you are applying (e.g. 'For continuity at x = 2, lim(x→2⁻) f(x) = lim(x→2⁺) f(x) = f(2)'), then compute each limit separately, and finally equate them to solve for the unknown. Allocate roughly 6 minutes per question.

Always write the continuity/differentiability condition explicitly
Compute left-hand limit, right-hand limit, and function value separately
Show algebraic steps: factorisation, L'Hôpital's rule, or standard limits
Box or underline your final numerical answer
Use NCERT Class 12 Mathematics standard notation: write lim(x→a) not lim x→a

Section E: Long Answer and HOTS Questions (5 marks each)

Long-answer questions demand multi-step reasoning, application of the chain rule to nested composite functions, or verification of both continuity and differentiability at a point. In CBSE 12 Mathematics board exams, these questions can carry up to 5 marks, so dedicate 10 minutes per question. Start by identifying what is given and what is to be proved or found. For differentiability questions, compute LHD and RHD using the first-principles definition: LHD = lim(h→0⁻) [f(a + h) – f(a)]/h. Show that LHD = RHD and conclude differentiability. For chain-rule problems involving inverse trigonometric functions, substitute smartly—e.g. let u = sin x, then express y in terms of u, differentiate, and multiply by du/dx. These questions often integrate algebra, trigonometry, and limits, making them excellent practice for the higher-order thinking skills tested in board papers.

Read the question twice and underline key phrases like 'prove', 'verify', 'find dy/dx'
Write 'To Prove' or 'Given' and 'To Find' as a preamble
Use substitution or the chain rule explicitly: state 'Let u = …, then y = … in terms of u'
For first-principles derivative, expand [f(a + h) – f(a)]/h and simplify before taking the limit
Check your final answer: does it have the correct domain? Is the sign consistent with the function's behaviour?

Case-Study Question (5 marks)

Case-study questions were introduced in the CBSE Class 12 Mathematics board exam pattern from 2021 onwards and typically carry 4–5 marks. They present a real-world scenario or a mathematical context followed by three to four sub-questions. For Continuity and Differentiability, a case study might describe a piecewise function modelling temperature variation, profit function, or motion of a particle, then ask you to determine points of discontinuity, verify differentiability, or compute the rate of change at a specific instant. Read the passage carefully, extract the piecewise definition or parameter values, and answer each sub-part methodically. These questions reward careful reading and the ability to apply NCERT Class 12 Mathematics concepts to slightly unfamiliar settings.

Read the case-study passage twice before attempting the sub-questions
Underline or note down the piecewise function definition and any given parameters
Each sub-question is usually worth 1 mark; write concise, direct answers
Use the concepts of continuity and differentiability exactly as you would in a standard question
Manage time: allocate no more than 8 minutes to the entire case study

Answer Key with Explanations

Below is the complete answer key for all sections. Each answer includes a brief explanation or working so you understand the reasoning. Use this key only after you have attempted the entire worksheet under timed conditions. Mark your answers honestly—give yourself 1 mark for each correct MCQ, fill-in-the-blank, or match/true-false item, 3 marks for each fully correct short-answer solution (award partial marks if your steps are correct but final answer is wrong), and 5 marks for each complete long-answer or case-study solution. A score above 70% indicates strong preparation; below 50% suggests you should revisit NCERT Class 12 Mathematics Chapter 5 notes and work through solved examples again. For personalised doubt-clearing and step-by-step photo solutions, consider using CBSETUTOR.ai—₹999 per month for all subjects, Classes 6–12, with a 3-day free trial.

How to Use This Worksheet for Maximum Benefit

Print two copies of this worksheet: one for a timed mock test and one for untimed review. On your first attempt, sit in a quiet room, set a 90-minute timer, and solve every question without referring to NCERT Class 12 Mathematics notes or Class 12 Mathematics solutions. This simulates board-exam pressure and highlights time-management issues. After the timer rings, stop immediately and mark your work using the answer key. For every mistake, write the correct solution in a different colour pen alongside your original attempt—this dual-colour method helps your brain register the error pattern. A day later, attempt the second copy untimed, focusing only on questions you got wrong or skipped. If you score below 60% on the timed attempt, dedicate an extra week to Chapter 5 revision, working through NCERT solved examples and additional CBSE 12 Mathematics practice papers. For instant doubt resolution and 24×7 homework help, parents in cities like Delhi, Bengaluru, and Mumbai are turning to CBSETUTOR.ai, which offers unlimited photo-upload question solving and AI-tutor chat for just ₹999 per month across all subjects and classes—try the 3-day free trial to see if it suits your child's learning style.

First attempt: timed, closed-book, 90 minutes—no NCERT help
Self-mark using the answer key; calculate your percentage score
Identify weak topics (e.g. chain rule, inverse trigonometric derivatives) and revise those sections in NCERT
Second attempt: untimed, focus on questions you got wrong or partially correct
Maintain an error log: write the question number, your mistake, and the correct reasoning
Repeat this cycle with additional worksheets every week during board exam preparation
Pair worksheet practice with daily 15-minute concept reviews from Class 12 Mathematics notes

Common Mistakes Students Make in Chapter 5 Continuity and Differentiability

Year after year, CBSE Class 12 Mathematics board papers reveal recurring errors. Many students forget that differentiability implies continuity but not vice versa; they incorrectly assume a continuous function must be differentiable. Another frequent slip is sign errors when computing left-hand and right-hand derivatives—always double-check the direction of h in the limit definition. In chain-rule problems involving inverse trigonometric functions, students often skip the substitution step and attempt direct differentiation, leading to algebraic chaos. For piecewise functions, forgetting to verify the function value f(a) separately from the left and right limits costs easy marks. Finally, time pressure in the exam causes students to write incomplete steps; even if your final answer is correct, you may lose a mark if the examiner cannot follow your logic. Practice writing every intermediate step clearly, as if explaining to a classmate who is seeing the problem for the first time.

Confusing 'continuous' with 'differentiable'—remember the |x| counter-example
Sign errors in LHD and RHD: check whether h approaches 0 from the left (h < 0) or right (h > 0)
Skipping substitution in chain-rule problems; always write 'Let u = …' explicitly
Not verifying f(a) when checking continuity; you need lim(x→a) f(x) = f(a)
Incomplete working: even correct answers lose marks if steps are missing in CBSE 12 Mathematics board exams
Misapplying L'Hôpital's rule outside 0/0 or ∞/∞ indeterminate forms
Forgetting domain restrictions for inverse trigonometric functions and logarithms

Why CBSETUTOR.ai Is the Smartest Revision Partner for Class 12 Mathematics

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Frequently asked questions

How many marks does Chapter 5 Continuity and Differentiability carry in CBSE Class 12 Mathematics board exam?+

Typically 8–10 marks across short-answer (3–4 marks) and long-answer (5–6 marks) questions. Some years include a case-study sub-question worth 1–2 marks from this chapter.

What is the difference between continuity and differentiability?+

A function is continuous at a point if the limit equals the function value there. It is differentiable if the derivative exists, which requires left-hand and right-hand derivatives to be equal. Every differentiable function is continuous, but not every continuous function is differentiable (e.g., f(x) = |x| at x = 0).

How do I check if a piecewise function is continuous at the point where the definition changes?+

Compute the left-hand limit, right-hand limit, and the function value at that point. If all three are equal, the function is continuous there. For differentiability, also check that left-hand derivative equals right-hand derivative.

Which formula sheet should I memorise for Continuity and Differentiability?+

Memorise derivatives of inverse trigonometric functions (sin⁻¹x, cos⁻¹x, tan⁻¹x, etc.), the chain rule, and standard limits like lim(x→0) sin x/x = 1. Keep the definition of LHD and RHD at your fingertips for first-principles questions.

Is it necessary to use L'Hôpital's rule for limits in Class 12 Mathematics?+

L'Hôpital's rule is not in the NCERT Class 12 syllabus; CBSE expects you to use algebraic simplification, factorisation, or standard limits. However, if you know L'Hôpital's rule and apply it correctly, you will not be penalised—just show clear steps.

How much time should I spend on this worksheet during revision?+

Allocate 90 minutes for a timed attempt under exam conditions, then another 30–45 minutes to review the answer key and rework mistakes. Repeat the cycle once a week for four weeks before the board exam.

Can I use a calculator for this worksheet?+

CBSE Class 12 Mathematics board exams do not permit calculators. Practice solving square roots, trigonometric values, and algebraic simplifications by hand. Use a calculator only for self-checking after you finish.

What are the most common errors in chain-rule questions?+

Students often forget to multiply by the derivative of the inner function. Always write 'Let u = inner function, then y = f(u)', differentiate y with respect to u, and multiply by du/dx. Missing this multiplication is the top reason for lost marks.

How does CBSETUTOR.ai help with Continuity and Differentiability doubts?+

Upload a photo of your problem—whether from this worksheet, NCERT exercises, or your school test—and receive a step-by-step solution within seconds. The AI tutor explains every algebraic step and highlights common pitfalls, making it ideal for late-night revision or quick concept checks.

Is this worksheet aligned with the latest CBSE 2025 exam pattern?+

Yes. It includes MCQs, fill-in-the-blanks, short-answer, long-answer, and a case-study question, mirroring the current CBSE Class 12 Mathematics question-paper structure. The difficulty distribution (20% easy, 50% moderate, 30% challenging) matches board-exam trends from recent years.

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