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Continuity and Differentiability — Revision Notes

JEE Mains · Mathematics

Free Continuity and Differentiability revision notes for JEE Mains Mathematics 2026 — key concepts, formulas, and definitions for quick revision.

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A Cartesian coordinate system showing the step-like graph of the signum function, f(x) = -1 for x<0, 0 for x=0, and 1 for x>0.
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Revision Notes — Continuity and Differentiability

Key concepts, formulas, and definitions from Continuity and Differentiability for JEE Mains Mathematics preparation.

Key Topics to Revise

1

Continuity of Functions

  • A function f(x) is continuous at x = a if lim(x→a) f(x) = f(a)
  • Three conditions for continuity: function is defined at the point, limit exists, and limit equals function value
  • Functions like polynomials, trigonometric, exponential, and logarithmic are continuous in their domains
2

Types of Discontinuities

  • Removable discontinuities: missing point or isolated point
  • Non-removable discontinuities: jump, infinite, or oscillatory type
  • Greatest integer function [x] has jump discontinuities at all integers
3

Differentiability

  • f'(a) = lim(h→0) [f(a+h) - f(a)]/h, provided limit exists and is finite
  • Left derivative f'(a⁻) and right derivative f'(a⁺) must be equal for differentiability
  • Differentiability implies continuity, but not vice versa
4

Continuity and Differentiability of Composite Functions

  • If f and g are continuous, then fog is continuous wherever defined
  • Chain rule: (fog)'(x) = f'(g(x)) · g'(x)
  • Composition can create or remove discontinuities

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Key Concepts

A function f(x) is continuous atDiscontinuities are classified asIf f(x) is continuous on [aA function f(x) is differentiable atIf f(x) is differentiable at x

Frequently Asked Questions

What topics are covered in Continuity and Differentiability for JEE Mains?
Continuity and Differentiability is an important chapter in JEE Mains Mathematics. It covers key concepts and formulas that are frequently tested in the exam. Key topics include: Continuity of Functions, Types of Discontinuities, Differentiability, Continuity and Differentiability of Composite Functions.
How important is Continuity and Differentiability for JEE Mains?
Continuity and Differentiability is a frequently tested chapter in JEE Mains Mathematics. Questions from this chapter appear regularly in previous year papers. There are 55 practice questions available for this chapter.
How to prepare Continuity and Differentiability for JEE Mains?
Start by understanding the core concepts, then solve practice questions. Focus on formulas and their applications. Use revision notes for quick review before the exam.

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