Continuity and Differentiability – JEE Mains Mathematics
1. Introduction
This chapter explores the core concepts of calculus — continuity and differentiability. It lays the foundation for understanding smooth behavior of functions and their derivatives.
2. Continuity
- A function f(x) is continuous at a point x = a if:
- f(a) is defined
- lim x→a f(x) exists
- lim x→a f(x) = f(a)
- Types of discontinuities: removable, jump, infinite.
- Polynomials, exponential, and trigonometric functions are continuous in their domain.
3. Differentiability
- If f′(x) exists, then f is differentiable at x.
- Differentiability implies continuity, but the converse is not always true.
- Check using left-hand and right-hand derivatives.
4. Algebra of Continuous and Differentiable Functions
- Sum, difference, product, and quotient of continuous/differentiable functions are also continuous/differentiable (where defined).
5. Chain Rule
- Used to differentiate composite functions.
If y = f(g(x)), then dy/dx = f′(g(x)) × g′(x)
6. Derivatives of Inverse Trigonometric Functions
- Standard derivatives include:
d/dx(sin⁻¹x) = 1 / √(1 - x²)
d/dx(cos⁻¹x) = -1 / √(1 - x²)
d/dx(tan⁻¹x) = 1 / (1 + x²)
7. Implicit Differentiation
- Used when y is not explicitly expressed in terms of x.
- Differentiate both sides and solve for dy/dx.
8. Logarithmic Differentiation
- Useful for differentiating complex expressions, especially with variables in powers.
- Take log on both sides, apply log rules, and then differentiate.
9. Mean Value Theorem
- Rolle’s Theorem: If a function is continuous and differentiable in [a, b], and f(a) = f(b), then there exists c ∈ (a, b) such that f′(c) = 0.
- Lagrange’s Mean Value Theorem: There exists c ∈ (a, b) such that
f′(c) = [f(b) – f(a)] / (b – a)