NCERT Class 12 Mathematics Mathematics Part-I: Chapter 5 — CONTINUITY AND DIFFERENTIABILITY
This chapter builds upon the differentiation concepts from Class XI, introducing the crucial ideas of continuity and differentiability for functions. It explores the relationship between these two concepts and introduces the differentiation of inverse trigonometric, exponential, and logarithmic functions. The chapter illustrates how differential calculus can represent geometrically intuitive conditions and presents fundamental theorems in this area. It begins by informally defining continuity through examples and graphs, highlighting functions that cannot be drawn without lifting the pen. A precise mathematical definition of continuity at a point 'c' is then provided: the limit of the function as x approaches 'c' must exist and be equal to the function's value at 'c'. This chapter is vital for understanding advanced calculus topics in CBSE Class 12 Mathematics.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-I |
| Chapter | Chapter 5 — CONTINUITY AND DIFFERENTIABILITY |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 732 |
Learning outcomes
- Understand the concept of continuity of a function at a point.
- Define continuity mathematically using limits.
- Identify points of discontinuity.
- Relate continuity and differentiability.
- Learn differentiation techniques for various function types.
Vocabulary
| Word | Meaning |
|---|---|
| Continuity | A function is continuous at a point if its graph can be drawn without lifting the pen. |
| Differentiability | The ability of a function to be differentiated at a point. |
| Limit | The value that a function approaches as the input approaches some value. |
| Left-hand limit | The value a function approaches as the input approaches from the left side. |
| Right-hand limit | The value a function approaches as the input approaches from the right side. |
| Discontinuity | A point where a function is not continuous. |
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Practice questions
- What is the informal definition of continuity at a point? Answer: A function is continuous at a point if its graph can be drawn around that point without lifting the pen from the paper.
- State the condition for a function f to be continuous at a point c. Answer: A function f is continuous at c if lim (x→c) f(x) = f(c).
- What is a point of discontinuity? Answer: A point where a function is not continuous is called a point of discontinuity.
Practice MCQs
Q1. A function f is continuous at a point c if:
Explanation: For a function to be continuous at a point c, the limit of the function as x approaches c must exist and be equal to the function's value at c.
Q2. In the function f(x) = 1 if x ≤ 0 and f(x) = 2 if x > 0, what is the left-hand limit at x = 0?
Explanation: As x approaches 0 from the left (x < 0), the function value is defined as 1.
Q3. If the left-hand limit and right-hand limit at a point do not coincide, the function is:
Explanation: For a limit to exist at a point, the left-hand and right-hand limits must be equal. If they are not, the function is discontinuous at that point.
Q4. Which of the following is an informal way to describe continuity?
Explanation: The intuitive idea of continuity is that the graph of the function can be drawn without any breaks, meaning without lifting the pen.
Frequently asked questions
What is the main focus of Chapter 5 of NCERT Class 12 Mathematics Part-I?
Chapter 5 focuses on the concepts of continuity and differentiability of functions, building upon previous knowledge of differentiation.
How is continuity defined mathematically in the chapter?
A function f is continuous at a point c if the limit of f(x) as x approaches c exists and is equal to f(c).
What does it mean for a function to be discontinuous at a point?
A function is discontinuous at a point if it is not continuous at that point, meaning the limit does not exist, the function is not defined, or the limit does not equal the function's value.
What types of functions will be differentiated in this chapter?
The chapter will cover differentiation of inverse trigonometric functions, exponential functions, and logarithmic functions.
What is the significance of continuity and differentiability in calculus?
Continuity and differentiability are fundamental concepts in calculus, essential for understanding rates of change, curve sketching, and advanced mathematical analysis.
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Topics covered
NCERT Class 12 Mathematics — Mathematics Part-I — Chapter 5 — CONTINUITY AND DIFFERENTIABILITY. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.