CBSE Class 12 Maths: Continuity & Differentiability — Notes 2026

Continuity of a Function

• f is continuous at x = a if: lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a)
• Discontinuity types: removable (limit exists but ≠ f(a)), jump (left ≠ right limit), infinite (limit = ∞)
• Theorem: if f and g are continuous at a, then f±g, fg, f/g (g≠0) are also continuous

Differentiability

• f is differentiable at x = a if LHD = RHD = finite value; LHD = lim(h→0⁻) [f(a+h)−f(a)]/h
• Differentiable → continuous but continuous ≠ differentiable (example: |x| continuous but not differentiable at x=0)
• f(x) = |x|: LHD = −1, RHD = +1 at x = 0; not differentiable (corner point)

Derivatives of Special Functions

• d/dx (log x) = 1/x; d/dx (eˣ) = eˣ; d/dx (aˣ) = aˣ ln a
• d/dx (sin⁻¹ x) = 1/√(1−x²); d/dx (cos⁻¹ x) = −1/√(1−x²); d/dx (tan⁻¹ x) = 1/(1+x²)
• Product rule: d(uv)/dx = u'v + uv'; Quotient rule: d(u/v)/dx = (u'v − uv')/v²

Chain Rule

• d/dx f(g(x)) = f'(g(x)) × g'(x); 'derivative of outer × derivative of inner'
• Example: d/dx sin(x²) = cos(x²) × 2x; d/dx e^(3x+1) = e^(3x+1) × 3
• Multiple applications: d/dx sin(cos(x²)) = cos(cos(x²)) × (−sin(x²)) × 2x

Implicit Differentiation

• When y is not explicitly defined as f(x); differentiate both sides w.r.t. x; collect dy/dx terms
• Example: x² + y² = 25 → 2x + 2y(dy/dx) = 0 → dy/dx = −x/y
• Trigonometric implicit: sin(x+y) = 1; use chain rule on left side

Logarithmic Differentiation

• Used when function has variable in both base and exponent: y = xˢⁱⁿˣ → ln y = sin x × ln x
• Differentiate: (1/y)(dy/dx) = cos x × ln x + sin x × (1/x); then multiply through by y
• Also useful for products of many functions: y = f(x)·g(x)·h(x) → take log, differentiate, multiply by y

CBSE Board Focus

• Continuity & Differentiability: 10–12 marks (largest chapter); chain rule, logarithmic differentiation, implicit most tested
• Check continuity: always evaluate both left and right limits AND function value at the point
• Parametric differentiation: x = f(t), y = g(t); dy/dx = (dy/dt)/(dx/dt) — frequently asked