---
type: "Chapter"
knowledge_type: "chapter"
entity_type: "chapter"
id: "66dfdf373f8b4e9e69bf7e4c"
title: "Continuity and Differentiability"
board: "CBSE"
curriculum: "CBSE"
class: "Class 12"
subject: "Mathematics"
book: "Mathematics Part - I"
chapter: "Continuity and Differentiability"
chapter_slug: "continuity-and-differentiability"
canonical_url: "https://www.edzy.ai/cbse-class-12-mathematics-mathematics-part-i-continuity-and-differentiability"
markdown_url: "https://www.edzy.ai/okf/chapter/cbse-class-12-mathematics-mathematics-part-i-continuity-and-differentiability.md"
source_type: "examSubjectBookChapter"
source_id: "66dfdf373f8b4e9e69bf7e4c"
source_pdf: "https://edzy-ai.s3.ap-south-1.amazonaws.com/edzy-express-ts/8564be4b-2e03-4183-b4ff-ac912ae80372.pdf"
source: "Edzy"
version: 1
last_updated: "2026-06-20"
---

# Continuity and Differentiability
This chapter explores continuity and differentiability of functions, introducing crucial concepts and techniques essential for calculus.

---

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part - I |
| Chapter | Continuity and Differentiability |
| Pages | 104-148 |

---

## Chapter Summary

### Short Summary
This chapter builds on previous knowledge from Class XI about differentiating polynomial and trigonometric functions and introduces concepts of continuity, differentiability, and inverse trigonometric function differentiation.

### Detailed Summary
We delve into the definitions and importance of continuity and differentiability. The chapter provides informal examples to illustrate continuity, leading to formal definitions. Additional examples demonstrate how to check for continuity at specific points and the relationship between limits and continuity. The importance of recognizing points of discontinuity is emphasized, alongside several types of functions and their continuity behaviors.

---

## Topic-Wise Explanation

### Introduction
This section continues the study of differentiation and its application, extending into continuity and differentiability.

### Continuity
Continuity is defined at a point using limits. A function \(f\) is continuous at a point \(c\) if:
$$\lim_{x 	o c} f(x) = f(c)$$. This section includes multiple examples illustrating how limits are used to determine continuity at various points.

### Differentiability
Differentiability is linked to the concept of continuity; a function must be continuous at a point to be differentiable there. The section further discusses methods to check differentiability.

### Exponential and Logarithmic Functions
New classes of functions, exponential and logarithmic, are introduced, highlighting their differentiation techniques and applications.

### Logarithmic Differentiation
This method simplifies the differentiation of products or quotients of functions, providing a powerful tool for complex functions.

### Derivatives of Functions in Parametric Forms
Attention is given to functions defined parametrically, explaining how to derive relationships between the variables involved.

### Second Order Derivative
The chapter discusses the significance of second derivatives in assessing concavity and points of inflection in functions.

---

## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Continuity | The concept of a function being continuous across its domain without breaks or jumps. |
| Differentiability | A function's smoothness and ability to have a derivative at a point. |

---

## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Left-hand limit | The value that a function approaches as the input approaches a specified point from the left. |
| Right-hand limit | The value that a function approaches as the input approaches a specified point from the right. |
| Point of discontinuity | A point where a function is not continuous. |

---

## Important Points for Revision

* A function is continuous at a point if left and right-hand limits equal the function's value at that point.
* Continuity across an interval requires points at both endpoints to be continuous.
* A function is discontinuous at a point if limits do not coincide with the function's value there.
* Polynomial and rational functions are continuous everywhere in their domain.
* A constant function is continuous at every point in its domain.
* The identity function is continuous across all real numbers.
* Functions defined piecewise must be analyzed for continuity at their breakpoints.
* Discontinuities can be classified as removable or non-removable.

---

## Practice Questions

### Short Answer Questions
1. Define continuity at a point.
2. Explain the significance of limits in determining continuity.
3. How can you check if a function is continuous at a point?
4. What is a point of discontinuity?
5. Give an example of a continuous function.

### Long Answer Questions
1. Discuss how to analyze the continuity of a piecewise function.
2. Explain the relationship between continuity and differentiability.
3. Provide examples of different functions and their behaviors regarding continuity across specific points.

---

## Source Attribution

| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 66dfdf373f8b4e9e69bf7e4c |
| Canonical URL | https://www.edzy.ai/cbse-class-12-mathematics-mathematics-part-i-continuity-and-differentiability |
| Markdown URL | https://www.edzy.ai/okf/chapter/cbse-class-12-mathematics-mathematics-part-i-continuity-and-differentiability.md |