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1.Continuity & Differentiability
Class 12 • Mathematics • NCERT
Continuity & Differentiability
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This chapter introduces the concepts of continuity and differentiability of functions, forming the foundation of differential calculus as prescribed by NCERT Class 12.
Lesson Objectives
- Understand the concept of continuity of a function.
- Learn differentiability and its conditions.
- Apply standard derivative formulas.
- Solve NCERT exam-based problems.
1. Continuity of a Function
A function f(x) is said to be continuous at x = a if:
(i) f(a) is defined
(ii) limx→a f(x) exists
(iii) limx→a f(x) = f(a)
(ii) limx→a f(x) exists
(iii) limx→a f(x) = f(a)
2. Left Hand Limit & Right Hand Limit
For continuity at a point, the left hand limit (LHL) and right hand limit (RHL) must exist and be equal.
limx→a⁻ f(x) = limx→a⁺ f(x)
3. Continuity in an Interval
A function is said to be continuous in an interval if it is continuous at every point of that interval.
Polynomials are continuous for all real x.
4. Differentiability
A function f(x) is differentiable at x = a if:
limh→0 [f(a + h) − f(a)] / h exists
5. Relation Between Continuity & Differentiability
Differentiability implies continuity, but continuity does not imply differentiability.
If a function is differentiable at a point, it must be continuous at that point.
6. Important NCERT Points
• Differentiability requires continuity
• Modulus functions may not be differentiable at some points
• Trigonometric functions are continuous in their domain
• Used extensively in applications of derivatives
• Modulus functions may not be differentiable at some points
• Trigonometric functions are continuous in their domain
• Used extensively in applications of derivatives
Practice Questions (NCERT)
- State the conditions for continuity at a point.
- Define differentiability.
- Is every continuous function differentiable?
- Is every differentiable function continuous?
- Are polynomial functions continuous?
- Write the definition of derivative using limits.
- What is LHL?
- What is RHL?
- At which point |x| is not differentiable?
- Does continuity ensure differentiability?
✅ Show Answer Key
- Existence of f(a), limit and equality
- Existence of derivative at a point
- No
- Yes
- Yes
- lim h→0 [f(a+h)−f(a)]/h
- Left Hand Limit
- Right Hand Limit
- x = 0
- No
© Aviate Learning – Continuity & Differentiability (NCERT Class 12)