Continuity and Differentiability Class 12: Key Results and Solved Examples

This chapter is short in theory but gives 4–5 marks of standard questions: check continuity at a point, find k, or apply Rolle’s/Lagrange’s theorem.

Continuity at a point

f is continuous at x = a if lim_x→a⁻ f(x) = lim_x→a⁺ f(x) = f(a). Polynomials, sin, cos, eˣ are continuous everywhere; rational functions where the denominator ≠ 0.

Find k type

f(x) = (x² − 9)/(x − 3) for x ≠ 3, k for x = 3. Limit = lim (x + 3) = 6 → k = 6.

Differentiability

• f′(a) exists if left-hand derivative = right-hand derivative.
• Differentiable ⇒ continuous; continuous does not imply differentiable (|x| at 0).
• Check corners, cusps and jumps: |x − 2| is not differentiable at 2.

Example

f(x) = x² for x ≤ 1, ax + b for x > 1; continuous and differentiable at 1. Continuity: 1 = a + b. Differentiability: 2 = a → a = 2, b = −1.

Mean value theorems

• Rolle: f continuous on [a, b], differentiable on (a, b), f(a) = f(b) ⇒ some c with f′(c) = 0
• Lagrange (MVT): f′(c) = (f(b) − f(a))/(b − a)
• Example: f(x) = x² − 4x + 3 on [1, 3]: f(1) = f(3) = 0, f′(c) = 2c − 4 = 0 → c = 2 ∈ (1, 3) ✓

Common mistakes

• Checking only the limit, not comparing with f(a)
• Assuming a function with a formula is always continuous (check denominators)
• Forgetting that c must lie strictly inside (a, b)

Related: Differentiation formulas.

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