Continuity and Differentiability

1. Introduction

Continuity and differentiability are foundational concepts in calculus. They describe how smoothly a function behaves and form the basis for understanding derivatives, integrals, and optimization.

2. Continuity

2.1 Definition

A function f is continuous at x = a if lim_{x→a} f(x) = f(a). This requires:

1. f(a) is defined.
2. lim_{x→a} f(x) exists (LHL = RHL).
3. lim_{x→a} f(x) = f(a).

2.2 Types of Discontinuity

Removable: The limit exists but does not equal f(a). Jump: LHL and RHL exist but are not equal. Infinite: One or both limits are infinite.

2.3 Continuity on an Interval

A function is continuous on [a, b] if it is continuous at every point in (a, b), right-continuous at a, and left-continuous at b.

3. Differentiability

3.1 Definition

f'(a) = lim_{h→0} (f(a + h) - f(a))/h, provided the limit exists.

'Differentiability implies continuity, but the converse is NOT true. Example: f(x) = |x| is continuous but not differentiable at x = 0.'

3.2 Standard Derivatives

d/dx(xⁿ) = nx^{n-1}, d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec² x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.

4. Chain Rule

If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.

'For composite functions, differentiate from outside to inside. Always verify you have differentiated every layer.'

5. Implicit Functions

If f(x, y) = 0 defines y implicitly, differentiate both sides with respect to x, treating y as a function of x, and solve for dy/dx.

6. Parametric Differentiation

If x = f(t) and y = g(t), then dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

7. Logarithmic Differentiation

For functions of the form y = [f(x)]^{g(x)}:

1. Take ln on both sides: ln y = g(x) ln f(x).
2. Differentiate: (1/y) dy/dx = g'(x) ln f(x) + g(x) f'(x)/f(x).
3. Multiply by y.

'Logarithmic differentiation is especially useful when the variable appears in both base and exponent.'

8. Second Order Derivatives

d²y/dx² = d/dx(dy/dx). In parametric form: d²y/dx² = (d/dt(dy/dx))/(dx/dt).

9. Mean Value Theorems

9.1 Rolle's Theorem

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) such that f'(c) = 0.

9.2 Lagrange's Mean Value Theorem (LMVT)

If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

10. Worked Problems

Problem 1: Check continuity of f(x) = {x² sin(1/x) if x ≠ 0, 0 if x = 0} at x = 0. Solution: lim_{x→0} x² sin(1/x) = 0 (bounded × 0). f(0) = 0. Hence continuous.

Problem 2: Find dy/dx if y = x^{sin x}. Solution: ln y = sin x · ln x. Differentiating: (1/y) dy/dx = cos x ln x + sin x/x. Therefore dy/dx = x^{sin x}(cos x ln x + sin x/x).

Problem 3: If x = a(θ - sin θ), y = a(1 - cos θ), find dy/dx. Solution: dx/dθ = a(1 - cos θ), dy/dθ = a sin θ. dy/dx = sin θ/(1 - cos θ) = cot(θ/2).

Problem 4: Verify Rolle's theorem for f(x) = x² - 4x + 3 on [1, 3]. Solution: f(1) = 0, f(3) = 0. f'(x) = 2x - 4 = 0 ⇒ x = 2 ∈ (1, 3). Verified.

11. Common Mistakes

'Students often forget to check all three conditions for continuity at a point. Missing the existence of limit is the most common error.'

'For implicit differentiation, remember to multiply by dy/dx every time you differentiate a term containing y.'

12. ISC Exam Focus

13. Self-Test Questions

1. Check the continuity of f(x) = |x - 1| + |x - 2| at x = 1 and x = 2.
2. Find dy/dx if x^y + y^x = a^b.
3. If y = (tan^{-1} x)², show that (1 + x²)² y₂ + 2x(1 + x²)y₁ = 2.
4. Verify LMVT for f(x) = x² - 3x + 2 on [1, 3].
5. Differentiate: y = (cos x)^{ln x} + (ln x)^{cos x}.

14. Quick Revision Summary

• Continuity requires LHL = RHL = f(a)
• Differentiability implies continuity (converse is false)
• Chain rule: dy/dx = dy/du × du/dx
• Implicit differentiation: differentiate every term, multiply by dy/dx for y-terms
• Parametric: dy/dx = (dy/dt)/(dx/dt)
• Logarithmic differentiation: take ln, differentiate, substitute back
• Rolle's: f(a) = f(b) ⇒ f'(c) = 0 for some c ∈ (a, b)
• LMVT: f'(c) = (f(b) - f(a))/(b - a)