Continuity and Differentiability

Board: GSEB Class: 12

Chapter Overview

A function f(x) is continuous at x = a if limit exists and equals f(a). Continuity on intervals, types of discontinuities (removable, jump, infinite). Differentiability implies continuity, but not vice versa. The chain rule, derivatives of implicit and parametric functions, logarithmic differentiation, Rolles Theorem, Mean Value Theorem.

Topics Covered

Continuity Definition
Types of Discontinuities
Differentiability
Chain Rule
Implicit Differentiation
Parametric Differentiation
Logarithmic Differentiation
Rolles Theorem
Mean Value Theorem

Key Formulas

f continuous at a: lim f(x) = f(a)

dy/dx = (dy/dt)/(dx/dt) for parametrics

ddx(f(g(x))) = f'(g(x))g'(x)

Real-World Applications

Applications: Motion analysis, function behavior analysis, optimization, error approximation.

Study Tips

Tip: Check continuity at boundary points carefully

Tip: Differentiability requires continuity first

Tip: Practice chain rule extensively