Continuity and Differentiability — Mathematics Class 12 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Continuity and Differentiability (Class 12) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Continuity and Differentiability (CBSE & HBSE)
Continuity and Differentiability is one of the largest chapters carrying 10-12 marks in CBSE. Topics include continuity checks, differentiation of composite/implicit/parametric functions, logarithmic differentiation, and mean value theorems.
Continuity at a Point and on an Interval
Continuity at a Point
A function f is continuous at x = c iff: lim(x→c⁻) f(x) = lim(x→c⁺) f(x) = f(c) (LHL = RHL = value of function)
Three conditions for continuity at x = c:
- f(c) is defined (f(c) exists)
- lim(x→c) f(x) exists (LHL = RHL)
- lim(x→c) f(x) = f(c)
If any ONE condition fails, f is discontinuous at x = c.
Types of Discontinuity
| Type | Description |
|---|---|
| Removable | LHL = RHL ≠ f(c), or f(c) undefined |
| Jump | LHL ≠ RHL (finite limits) |
| Infinite | At least one limit is ∞ |
Algebra of Continuous Functions
If f and g are continuous at x = c, then:
- f+g, f−g, fg are continuous at x = c
- f/g is continuous at x = c (if g(c) ≠ 0)
- |f|, f∘g (composite) are continuous at x = c
Standard Continuous Functions
- Polynomial functions: continuous everywhere
- Rational functions: continuous wherever denominator ≠ 0
- Trigonometric functions: sin, cos continuous everywhere; tan continuous except at x = (2n+1)π/2
- Exponential and logarithmic functions: continuous on their domains
Piecewise functions: For f defined piecewise at x=a, compute LHL, RHL, and f(a) separately.
CBSE Tip: CBSE regularly asks to find values of constants a, b in piecewise-defined functions that make them continuous. Set LHL = RHL = f(a) and solve simultaneously.
Differentiability and Differentiation Techniques
Differentiability
Definition: f is differentiable at x = c iff: f'(c) = lim(h→0) [f(c+h) − f(c)] / h exists finitely.
Relationship: Differentiability ⟹ Continuity, but NOT vice versa. |x| is continuous but NOT differentiable at x = 0 (left derivative = −1, right derivative = +1).
Standard Derivatives
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| sin⁻¹x | 1/√(1−x²) |
| cos⁻¹x | −1/√(1−x²) |
| tan⁻¹x | 1/(1+x²) |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Example: d/dx[sin(x²)] = cos(x²) · 2x
Implicit Differentiation
For f(x,y) = 0, differentiate both sides w.r.t. x, treating y as a function of x: dy/dx = −(∂f/∂x)/(∂f/∂y)
Example: x² + y² = r² ⟹ 2x + 2y(dy/dx) = 0 ⟹ dy/dx = −x/y
Logarithmic Differentiation
Used when: function is of the form [f(x)]^g(x) Step 1: Take log of both sides: ln y = g(x)·ln f(x) Step 2: Differentiate: (1/y)(dy/dx) = g'(x)·ln f(x) + g(x)·f'(x)/f(x) Step 3: Multiply by y
Parametric Differentiation
If x = f(t) and y = g(t): dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t)
CBSE Tip: Logarithmic differentiation is tested every year for expressions like xˣ, (sin x)ˢⁱⁿˣ, or products of several functions. Always take log first.
Second Order Derivatives and Mean Value Theorems
Second Order Derivative
d²y/dx² = d/dx(dy/dx)
Denoted as y₂, f''(x), or D²y.
Example: y = x³ ⟹ dy/dx = 3x² ⟹ d²y/dx² = 6x
For parametric equations: If x=f(t), y=g(t): d²y/dx² = [d/dt(dy/dx)] / (dx/dt)
Rolle's Theorem
Statement: If f satisfies:
- f is continuous on [a, b]
- f is differentiable on (a, b)
- f(a) = f(b)
Then ∃ at least one c ∈ (a,b) such that f'(c) = 0.
Geometric interpretation: There is at least one point where the tangent is horizontal.
Lagrange's Mean Value Theorem (LMVT)
Statement: If f is:
- Continuous on [a, b]
- Differentiable on (a, b)
Then ∃ at least one c ∈ (a,b) such that: f'(c) = [f(b) − f(a)] / (b − a)
Geometric interpretation: There is at least one point where the tangent slope equals the slope of the chord AB.
Note: Rolle's Theorem is a special case of LMVT when f(a) = f(b).
Applying LMVT — Steps
- Verify continuity on [a,b]
- Verify differentiability on (a,b)
- Compute f'(c) = [f(b)−f(a)]/(b−a)
- Solve for c and verify c ∈ (a,b)
CBSE Tip: In LMVT questions, ALWAYS verify both conditions before finding c. Many students skip the verification and lose marks.
Frequently asked questions
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Do these notes follow CBSE and HBSE?
Yes. The Continuity and Differentiability notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Continuity and Differentiability chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Continuity and Differentiability.