This page hosts the Continuity and Differentiability Class 12 Notes, presenting it for Class 12 Mathematics Chapter 5 Continuity and Differentiability. The the resource are sectioned to match the NCERT chapter, with each section closing in a short summary box and a single solved example. Suitable for both board and JEE Main preparation, the notes are free to download.
- CBSE Class 12 Boards: 8 to 10 marks (largest single-chapter weightage in the Calculus unit)
- JEE Main: 3 to 4 questions per sitting, roughly 7 to 9% of the Mathematics section
- CUET (UG) Maths: 2 to 3 direct MCQs on continuity tests, chain rule, and logarithmic differentiation each year
Student Pulse - Continuity and Differentiability Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
These Collegedunia the chapter notes are compiled by Class 12 Maths specialists and cross-verified against the 2026-27 NCERT textbook and the past five years of CBSE marking schemes, covering continuity tests, all 16 standard derivatives, logarithmic differentiation, parametric and second-order derivatives, and the two mean value theorems.
Also Check:
- NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability
- Class 12 Maths Chapter 5 Continuity and Differentiability Formula Sheet
- Class 12 Maths Chapter 5 Continuity and Differentiability Handwritten Notes
Why Continuity and Differentiability Matters for Class 12 Boards and JEE
This chapter is the gatekeeper of the Calculus unit. A student shaky on the chain rule or on Rolle's / MVT typically loses 12 to 15 marks across the Calculus unit, far more than the the PDF's own 10-mark allotment. CBSE 2024 and 2025 both carried a 5-mark question requiring logarithmic differentiation of y = [f(x)]g(x) .
Continuity and Differentiability Video Walkthrough
Source: NCERT Wallah on YouTube
How will Collegedunia's NCERT Notes for Class 12 Maths Chapter 5 help you?
- The three-line continuity test is templated so you write it identically every time, earning method marks even when the algebra is shaky.
- Derivatives of all 16 standard functions are listed in one table, the format JEE Main rewards for speed.
- Logarithmic differentiation is shown as a 4-step procedure for y = [f(x)]g(x) , the recurring 5-mark board question.
- Parametric and second-order derivatives are treated together, since CBSE pairs them in long answers almost every year.
- Rolle's Theorem and the Mean Value Theorem are written in CBSE's preferred verification format (state the three hypotheses, check each, find c ).
Prerequisite Map: What to Revise Before Chapter 5
- Limits and one-sided limits (Class 11 Ch 13): the basis of every continuity test.
- sin x / x limit: x → 0 sin xx = 1 is used to derive trigonometric derivatives.
- Inverse trigonometric domains (Ch 2): derivatives of sin-1 x , cos-1 x , tan-1 x all carry domain restrictions.
- Laws of logarithms: log(ab) = b log a and log(ab) = log a + log b drive logarithmic differentiation.
Class 12 Maths Chapter 5 Topic-by-Topic Breakdown
1. Continuity at a point
f is continuous at x = a if f(a) is defined, x → a f(x) exists, and the two agree: $$ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) $$ Any one failure breaks continuity. Easy Tip: write the three-line test verbatim; CBSE awards a method mark just for stating it.
2. Algebra of continuous functions
If f, g are continuous at a , so are f ± g , fg , and f/g (with g(a) ≠ 0 ). Composition preserves continuity, which is how polynomials, sine, cosine, exponential, and logarithm are continuous on their domains without point-wise checks.
3. Differentiability at a point
f is differentiable at a if $$ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} $$ exists, i.e. left- and right-hand derivatives agree. Theorem: differentiable implies continuous, not conversely. |x| is continuous at 0 but not differentiable there: left derivative -1 , right +1 .
4. Derivatives of standard functions
The 16 standard derivatives drive 70 percent of every question in the this chapter.
| Function | Derivative |
|---|---|
| xn | n xn-1 |
| sin x , cos x | cos x , -sin x |
| tan x , cot x | sec2 x , -csc2 x |
| sec x , csc x | sec x tan x , -csc x cot x |
| sin-1 x , cos-1 x | ± 1√1 - x2 , |x| < 1 |
| tan-1 x , cot-1 x | ± 11 + x2 |
| ex , ax | ex , ax log a |
| log x , a x | 1x , 1x log a |
Domain qualifiers matter: CBSE deducts a method mark whenever a student omits the restriction on inverse-trig or log derivatives.
5. Chain rule
If y = f(u) , u = g(x) , then dydx = f'(g(x)) · g'(x) . For y = sin(x2) : dydx = 2x cos(x2) . Common Confusion: stopping after the outer derivative; always multiply by g'(x) .
6. Implicit differentiation
Differentiate every term in F(x, y) = 0 with respect to x , treating y as a function of x . For x2 + y2 = 25 : dydx = -xy .
7. Logarithmic differentiation
Use it for y = [f(x)]g(x) or long products. Four steps: (i) log y = g(x) log f(x) ; (ii) differentiate, product rule on the right; (iii) the left gives 1y dydx ; (iv) multiply by y , substitute back.
For y = xx : dydx = xx (log x + 1) . This setup is a 5-mark question in 4 of the last 6 CBSE papers.
8. Parametric and second-order derivatives
If x = f(t) , y = g(t) , then dydx = g'(t)f'(t) . The second derivative is d2 ydx2 = ddx(dydx) .
Board Tip: to prove (1 - x2) y2 - x y1 = 0 for y = sin-1 x , differentiate √1 - x2 · y1 = 1 rather than computing y2 directly.
9. Rolle's Theorem and the Mean Value Theorem
Rolle's: if f is continuous on [a, b] , differentiable on (a, b) , and f(a) = f(b) , then f'(c) = 0 for some c ∈ (a, b) . MVT (Lagrange): drop the third hypothesis; then
$$ f'(c) = \frac{f(b) - f(a)}{b - a} $$
for some c ∈ (a, b) . CBSE awards 2 of the 4 marks just for stating and verifying the hypotheses; never skip straight to solving for c .
Continuity and Differentiability Important Derivations for Class 12 Boards
- ddx(sin x) = cos x from first principles using h → 0 sin hh = 1 .
- ddx(sin-1 x) = 1√1 - x2 via implicit differentiation of sin y = x .
- ddx(log x) = 1x from first principles using h → 0 log(1 + h/x)h/x = 1 .
- ddx(ex) = ex from first principles using h → 0 eh - 1h = 1 .
- Differentiability implies continuity: if f'(a) exists, then x → a f(x) = f(a) .
- Verification of Rolle's / MVT for a given function on [a, b] , the templated 4-mark question.
Common Misconceptions in Continuity and Differentiability
- "Continuous implies differentiable." The converse is true; |x| is continuous at 0 but has a corner there.
- Wrong power rule for variable exponents: ddx[f(x)]g(x) needs logarithmic differentiation.
- Skipping domain restrictions on inverse-trig derivatives, e.g. omitting |x| < 1 for sin-1 x .
- Applying Rolle's when f(a) ≠ f(b) : the third hypothesis fails; switch to the Mean Value Theorem.
Continuity and Differentiability Class 12: Most Important Topics
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| NCERT Class 12 Maths Chapter 5 Notes: Important Topics | |
|---|---|
| Continuity at a point and on an interval | Algebra of continuous functions |
| Differentiability at a point | Differentiability implies continuity |
| Derivatives of 16 standard functions | Chain rule for compositions |
| Derivatives of inverse trigonometric functions | Exponential and logarithmic derivatives |
| Logarithmic differentiation y = [f(x)]g(x) | Implicit differentiation |
| Parametric form derivatives | Second-order derivatives |
| Rolle's Theorem (verification) | Lagrange's Mean Value Theorem |
Treat the table as your tick-list during revision: every row maps to at least one CBSE question type from the last five papers.
Most Repeated Questions in CBSE Class 12 Boards: Chapter 5
Ques. If f(x) = sin 3xx for x ≠ 0 and f(0) = k is continuous at 0, find k . (2025, 2023, 2020)
[3-Mark] x → 0 sin 3xx = 3 . Hence k = 3 .
Ques. Differentiate xx + xsin x . (2024, 2022, 2019)
[5-Mark] ddx(xx) = xx(log x + 1) and ddx(xsin x) = xsin x(cos x log x + sin xx) . Add the two.
Ques. If x = a(θ - sinθ) , y = a(1 - cosθ) , find d2 ydx2 at θ = π/2 . (2024, 2021)
[5-Mark] dydx = cot(θ/2) . Then d2 ydx2 = ddθ[cot(θ/2)] · dθdx , evaluated at π/2 .
Ques. Verify Rolle's Theorem for f(x) = x2 - 4x + 3 on [1, 3] . (2023, 2020)
[4-Mark] Polynomial, hence continuous on [1, 3] and differentiable on (1, 3) ; f(1) = 0 = f(3) . Solve f'(c) = 2c - 4 = 0 : c = 2 ∈ (1, 3) .
Continuity and Differentiability Class 12 Notes: Previous Year Paper Analysis
Across the 2025 to 2020 CBSE papers, Chapter 5 has settled into a predictable mark distribution.
- Every paper carries at least one MCQ on the three-line continuity test, often with a piecewise function as in 2025.
- Logarithmic differentiation of y = [f(x)]g(x) appeared as a 5-mark question in 4 of the last 6 papers.
- Parametric or second-order derivatives is the next-most-frequent 5-marker.
- Rolle's or MVT verification appears almost every alternate year as a 4-mark question.
Full year-wise PYQ map: NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability
Weightage of Continuity and Differentiability in JEE Main, CUET and CBSE 2026
Beyond the boards, Chapter 5 is the doorway to the entire Calculus block in JEE Main. The chapter accounts for nearly one in every twelve Maths questions across recent JEE Main sittings.
| Exam | Weightage | Important Topics |
|---|---|---|
| JEE Main 2025 | 3 to 4 questions (~7 to 9%) | Chain rule, logarithmic differentiation, MVT proofs, continuity counter-examples |
| CBSE Boards 2025 | 8 to 10 marks | Three-line continuity test, logarithmic diff., parametric second-order, Rolle's verification |
| CUET (UG) 2025 Maths | 2 to 3 MCQs | Standard derivatives, chain rule, basic continuity checks |
| JEE Main 2024 | 3 to 4 questions (~7%) | Differentiability of |x| -like functions, MVT numericals |
| JEE Main 2026 | Pending (exam rescheduled) | - |
The pattern is steady: CBSE has never skipped these notes in the last six years, and JEE Main recycles the chain rule plus logarithmic differentiation as a stems-multiplier.
NCERT Notes for Class 12 Maths: All Chapters
Jump to any other Class 12 Maths chapter's Notes page. Ch 5 is excluded.
| Chapter | Chapter Notes |
|---|---|
| Chapter 1 | Relations and Functions Notes |
| Chapter 2 | Inverse Trigonometric Functions Notes |
| Chapter 3 | Matrices Notes |
| Chapter 4 | Determinants Notes |
| Chapter 6 | Application of Derivatives Notes |
| Chapter 7 | Integrals Notes |
| Chapter 8 | Application of Integrals Notes |
| Chapter 9 | Differential Equations Notes |
| Chapter 10 | Vector Algebra Notes |
| Chapter 11 | Three Dimensional Geometry Notes |
| Chapter 12 | Linear Programming Notes |
| Chapter 13 | Probability Notes |
this Class 12 page: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Continuity and Differentiability Class 12 Notes - Quick Summary
- The the resource cover every section of Class 12 Mathematics Chapter 5 Continuity and Differentiability, aligned to the 2026-27 NCERT print.
- The chapter notes include formal definitions, solved examples and end-of-section formula recap suitable for board and JEE Main preparation.
- The the PDF are downloadable as a free PDF and follow the notation of the official NCERT textbook line for line.
Exercise-wise Breakdown of the Continuity and Differentiability Chapter
The Continuity and Differentiability chapter splits into 7 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 5.1 | Continuity at a point and on an interval |
| Exercise 5.2 | Algebra of continuous functions |
| Exercise 5.3 | Differentiability and chain rule |
| Exercise 5.4 | Derivatives of inverse trigonometric functions |
| Exercise 5.5 | Logarithmic differentiation |
| Exercise 5.6 | Parametric and implicit differentiation |
| Exercise 5.7 | Second-order derivatives; Rolle's and Mean Value Theorem |
| Miscellaneous Exercise | Mixed continuity and differentiability problems |
PDF Download Formats and Languages for the Continuity and Differentiability Chapter
The Continuity and Differentiability Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The continuity and differentiability class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the continuity and differentiability class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The continuity and differentiability class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The continuity and differentiability class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The continuity and differentiability class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the continuity and differentiability class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Continuity and Differentiability Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Continuity and Differentiability chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable continuity and differentiability class 12 important questions you will see on board day.
- continuity and differentiability class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The continuity and differentiability class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching continuity and differentiability class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Continuity and Differentiability chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full continuity and differentiability class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Continuity and Differentiability Notes Pair with NCERT Solutions and the Formula Sheet
The Continuity and Differentiability Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Continuity and Differentiability Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| continuity and differentiability class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| continuity and differentiability class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The continuity and differentiability class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The continuity and differentiability class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The continuity and differentiability class 12 formulas reference sheet is the same A4 file students sometimes refer to as continuity and differentiability class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board continuity and differentiability class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 continuity and differentiability solutions, class 12 continuity and differentiability ncert solutions, ncert class 12 continuity and differentiability solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to Continuity and Differentiability Class 12 |
|---|---|
| RD Sharma Class 12 Continuity and Differentiability | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Continuity and Differentiability | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the PDF | Free online walkthroughs; useful for video-style learning. |
| Shaalaa continuity and differentiability class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board this chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Continuity and Differentiability | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Continuity and Differentiability Notes Page Most Effectively
The recommended study plan for these notes chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the continuity and differentiability class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The continuity and differentiability class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Continuity and Differentiability Class 12 Notes - Frequently Asked Questions
Ques. What is the weightage of Continuity and Differentiability in CBSE Class 12 Maths Board Exam 2026?
Ans. Continuity and Differentiability typically carries 8 to 10 marks in the CBSE Class 12 Maths paper, the largest single-chapter weightage in the Calculus unit. Expect at least one MCQ on continuity of a piecewise function, a 3-mark short answer on finding a constant for continuity, and a 5-mark long answer that is almost always either logarithmic differentiation or a parametric second-order derivative.
Ques. Is every continuous function differentiable?
Ans. No. Differentiability implies continuity, but the converse is false. The classic counter-example is f(x) = |x| at x = 0 , which is continuous but not differentiable because the left-hand derivative is -1 and the right-hand derivative is +1 . A continuous function with a corner, cusp, or vertical tangent fails differentiability at that point.
Ques. When should I use logarithmic differentiation?
Ans. Use logarithmic differentiation in two situations: (i) when the function has the form y = [f(x)]g(x) , where both base and exponent are functions of x ; and (ii) when the function is a product or quotient of many factors, so taking log converts the product into a sum.
The 4-step procedure (log both sides, differentiate, isolate dydx , substitute back) is standard.
Ques. How do I verify Rolle's Theorem for a given function on an interval?
Ans. State the three hypotheses and check each one explicitly: (i) f is continuous on the closed interval [a, b] , (ii) differentiable on the open interval (a, b) , and (iii) f(a) = f(b) .
If any hypothesis fails, say so and stop. If all three hold, solve f'(c) = 0 for c and confirm c ∈ (a, b) . CBSE awards 2 of the 4 marks just for the verification.
Ques. What is the chain rule and where do I apply it?
Ans. The chain rule is dydx = dydu · dudx for a composed function y = f(g(x)) . Apply it whenever one function is wrapped inside another, for instance y = sin(x2) , y = etan x , or y = log(sin x) . For multiple nesting, apply the rule layer by layer.
Ques. Why are domain restrictions on inverse trigonometric derivatives important?
Ans. The derivatives of sin-1 x and cos-1 x are valid only for |x| < 1 ; at x = ± 1 the tangent line is vertical and the derivative does not exist. Examiners deduct a method mark whenever a student writes the derivative without flagging the restriction.
Ques. How is Chapter 5 connected to later chapters in the Calculus unit?
Ans. Every later Calculus chapter rests on Continuity and Differentiability. Application of Derivatives (Ch 6) uses derivatives for tangents, normals, and extrema. Integrals (Ch 7) is the inverse operation. Differential Equations (Ch 9) builds on second-order derivatives. Treat Chapter 5 as these notes that pays off across the rest of the syllabus.
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