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Continuity and Differentiability – Class 12 Maths Chapter 5 Notes & NCERT PDF

Class 12MathematicsChapter 5NCERT book: Mathematics Part I - Class 12

Notes and a simple summary of Chapter 5 of the NCERT Class 12 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 5: Continuity and Differentiability

Introduction

Class XI introduced derivatives of polynomials and trigonometric functions. This chapter builds the two foundations of calculus — continuity and differentiability — and extends differentiation to composite, implicit, inverse trigonometric, exponential, logarithmic and parametric functions, ending with second order derivatives. Loosely, a function is continuous if its graph can be drawn without lifting the pen.

Key Concepts & Theorems

5.2 Continuity

  • Definition: f is continuous at x = c if lim(x→c) f(x) = f(c), i.e. left-hand limit = right-hand limit = value of the function at c.
  • f is continuous (on its domain) if it is continuous at every point of its domain.
  • Examples from the book:

    • - f(x) = 2x + 3 is continuous at x = 1 (limit 5 = f(1)).

    - f(x) = |x| is continuous at 0 and everywhere.

    - Every polynomial and every constant function is continuous.

    - The greatest integer function [x] is discontinuous at every integer.

  • 5.2.1 Algebra of continuous functions (Theorem 1): if f and g are continuous at c, then f + g, f − g, f·g and f/g (with g(c) ≠ 0) are continuous at c. Hence every rational function is continuous on its domain, and sin x, cos x, tan x are continuous.
  • Theorem 2: a composition of continuous functions is continuous, e.g. sin(x²).

    • 5.3 Differentiability

  • f is differentiable at c if the derivative f′(c) = lim(h→0) [f(c + h) − f(c)]/h exists, i.e. left-hand derivative = right-hand derivative.
  • Theorem 3: every differentiable function is continuous. The converse is false: |x| is continuous at 0 but not differentiable there (LHD = −1, RHD = +1).
  • 5.3.1 Chain rule (Theorem 4): if y = f(t) and t = u(x), then dy/dx = (dy/dt)(dt/dx). Example: d/dx sin(x²) = 2x cos(x²).
  • 5.3.2 Implicit functions: differentiate both sides w.r.t. x, treating y as a function of x, then solve for dy/dx.
  • 5.3.3 Inverse trigonometric functions:

    • - d/dx sin⁻¹x = 1/√(1 − x²)

    - d/dx cos⁻¹x = −1/√(1 − x²)

    - d/dx tan⁻¹x = 1/(1 + x²)

    5.4 Exponential and logarithmic functions

  • d/dx eˣ = eˣ; d/dx log x = 1/x (x > 0); d/dx aˣ = aˣ log a.
  • e^(log x) = x only for x > 0, since log x is defined only for positive x.

    • 5.5 Logarithmic differentiation

    Used for functions of the form [u(x)]^v(x) or long products and quotients: take log of both sides, then differentiate.

  • Example 29: y = x^(sin x) ⇒ log y = sin x · log x ⇒ dy/dx = x^(sin x) (cos x · log x + sin x / x).

    • 5.6 Parametric forms

    If x = f(t) and y = g(t), then dy/dx = (dy/dt)/(dx/dt), with dx/dt ≠ 0. Example: x = a cos θ, y = a sin θ ⇒ dy/dx = −cot θ.

    5.7 Second order derivative

    d²y/dx² = d/dx(dy/dx), also written y″ or f″(x).

    Worked Examples

  • Continuity of |x| at 0 (Example 3). LHL = RHL = 0 = f(0), so |x| is continuous at 0.
  • Chain rule (Example 21). f(x) = sin(x²): with t = x², f′(x) = cos t · 2x = 2x cos(x²).
  • Inverse trig (Example 24). y = sin⁻¹x ⇒ sin y = x ⇒ cos y · dy/dx = 1 ⇒ dy/dx = 1/cos y = 1/√(1 − x²).
  • Exponential (Example 28). aˣ = e^(x log a) ⇒ d/dx aˣ = e^(x log a) · log a = aˣ log a.
  • Second order (Example 36). y = A sin x + B cos x ⇒ y″ = −A sin x − B cos x = −y, so y″ + y = 0.
  • Second order (Example 37). y = 3e²ˣ + 2e³ˣ ⇒ y′ = 6e²ˣ + 6e³ˣ, y″ = 12e²ˣ + 18e³ˣ ⇒ y″ − 5y′ + 6y = 0.
  • Implicit differentiation. For the circle x² + y² = 25, differentiate term by term: 2x + 2y · dy/dx = 0, so dy/dx = −x/y. At the point (3, 4) the slope is −3/4.
  • Logarithmic differentiation. y = xˣ (x > 0): log y = x log x ⇒ (1/y) dy/dx = log x + 1 ⇒ dy/dx = xˣ(1 + log x).

    • How to choose the right technique

    Before differentiating, read the function carefully. A function inside another function calls for the chain rule. An equation mixing x and y that cannot easily be solved for y calls for implicit differentiation. A variable in both the base and the exponent, or a long product or quotient, calls for logarithmic differentiation. When both x and y are given in terms of a third variable, use the parametric formula. Many board questions combine two techniques, so practise identifying them quickly.

    Formula Sheet

  • Continuity at c: lim(x→c) f(x) = f(c)
  • (u ± v)′ = u′ ± v′; (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²
  • Chain rule: dy/dx = dy/dt · dt/dx
  • d/dx sin⁻¹x = 1/√(1 − x²); d/dx cos⁻¹x = −1/√(1 − x²); d/dx tan⁻¹x = 1/(1 + x²)
  • d/dx eˣ = eˣ; d/dx log x = 1/x; d/dx aˣ = aˣ log a
  • Parametric: dy/dx = (dy/dt)/(dx/dt)
  • Differentiable ⇒ continuous (not conversely)

    • Board Exam Focus

  • 1–2 marks: find k so that a piecewise function is continuous at a point; differentiate a composite function; derivative of an inverse trig function.
  • 3 marks: logarithmic differentiation (xˣ, x^(sin x)); implicit differentiation; parametric dy/dx.
  • 4–5 marks: prove a relation involving the second derivative (like y″ + y = 0); continuity and differentiability of a piecewise function at the joining point.
  • Outline for "find k" questions: (i) compute LHL, (ii) compute RHL, (iii) compute f(c), (iv) equate and solve for k, (v) state the value.

    • Key Terms

  • Continuity (सांतत्य)
  • Continuous function (संतत फलन)
  • Discontinuity (असांतत्य)
  • Differentiability (अवकलनीयता)
  • Derivative (अवकलज)
  • Left-hand limit (बाएँ पक्ष की सीमा)
  • Right-hand limit (दाएँ पक्ष की सीमा)
  • Chain rule (शृंखला नियम)
  • Implicit function (अस्पष्ट फलन)
  • Composite function (संयुक्त फलन)
  • Logarithmic differentiation (लघुगणकीय अवकलन)
  • Parametric form (प्राचलिक रूप)
  • Second order derivative (द्वितीय कोटि का अवकलज)
  • Greatest integer function (महत्तम पूर्णांक फलन)

    • Common Mistakes

  • Assuming continuity implies differentiability — |x| at 0 is the standard counter-example.
  • Forgetting the inner derivative in the chain rule (writing cos(x²) for d/dx sin(x²)).
  • Differentiating xˣ as x · xˣ⁻¹ or as xˣ log x — neither is right; use logarithms: d/dx xˣ = xˣ(1 + log x).
  • In implicit differentiation, forgetting to multiply by dy/dx when differentiating a term in y.
  • Missing the minus sign in d/dx cos⁻¹x.
  • Checking only one side (LHL or RHL) when testing continuity of a piecewise function.

    • 💡 Key Learning Points

    • ✓Test continuity at a point using lim(x→c) f(x) = f(c), comparing left-hand limit, right-hand limit and f(c).
    • ✓Know that differentiability implies continuity but not conversely, with |x| at 0 as the standard counter-example.
    • ✓Differentiate composite, implicit and inverse trigonometric functions using the chain rule.
    • ✓Differentiate exponential, logarithmic and [u(x)]^v(x) functions, using logarithmic differentiation where needed.
    • ✓Find dy/dx for parametric functions and compute second order derivatives to prove relations such as y″ + y = 0.

    👨‍🏫 Teaching Tips

    • →Start with graphs: ask students which curves can be drawn without lifting the pen, then formalise with limits.
    • →Use a table of LHL, RHL and f(c) for every piecewise function so the three checks are never skipped.
    • →Show the corner of y = |x| at the origin and compute LHD = −1, RHD = +1 to make non-differentiability concrete.
    • →Teach the chain rule as 'outside derivative × inside derivative' with colour-coding of the inner function.
    • →Derive d/dx sin⁻¹x in class from sin y = x rather than just stating it.
    • →Give a mixed worksheet where students first decide which technique (chain, implicit, log, parametric) to use.
    • →End each lesson with a short 'prove y″ + … = 0' problem to build fluency with second derivatives.

    📋 Assessment Questions

    1. Can the student find k so that a piecewise function is continuous at a given point?
    2. Can the student explain why |x| is continuous but not differentiable at 0?
    3. Can the student apply the chain rule and logarithmic differentiation correctly?
    4. Can the student find dy/dx in parametric form and verify a relation using the second derivative?

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    Keep learning

    All chapters in Mathematics Part I - Class 12

    1. 1. Relations and Functions
    2. 2. Inverse Trigonometric Functions
    3. 3. Matrices
    4. 4. Determinants
    5. 5. Continuity and Differentiability
    6. 6. Application of Derivatives