Mathematics Class 12 - Continuity And-Differentiability Notes

Continuity and Differentiability

Welcome to the chapter on Continuity and Differentiability for Class 12. In this chapter, you will learn about the concepts of continuity and differentiability of functions, how to check them, and their applications in calculus. By the end of this chapter, you will be able to solve problems involving continuity, differentiability, and understand the relationship between them.

Key Concepts

• Continuity: A function is continuous at a point if its left-hand limit, right-hand limit, and value at that point are all equal.
• Differentiability: A function is differentiable at a point if its derivative exists at that point.
• Relationship: Every differentiable function is continuous, but every continuous function may not be differentiable.

Continuity of a Function

A function f(x) is continuous at x = a if:

• Left-hand limit (LHL) at a = Right-hand limit (RHL) at a = f(a)

Example: Check the continuity of f(x) = x² at x = 2.
LHL = RHL = f(2) = 4, so the function is continuous at x = 2.

Differentiability of a Function

A function f(x) is differentiable at x = a if the left-hand derivative and right-hand derivative at a are equal.

• If f'(a^-) = f'(a⁺), then f(x) is differentiable at x = a.

Example: f(x) = x² is differentiable everywhere because its derivative exists at all points.

Important Results

• If a function is differentiable at a point, it is also continuous at that point.
• A function can be continuous but not differentiable (e.g., f(x) = |x| at x = 0).

Practice Questions

1. Check the continuity of f(x) = 3x + 2 at x = 1.
2. Is f(x) = |x| differentiable at x = 0? Justify your answer.
3. If f(x) = x³, find f'(x) and check its continuity and differentiability at x = 0.
4. Give an example of a function that is continuous but not differentiable at a point.
5. State the relationship between continuity and differentiability.

Challenge Yourself

• Prove that f(x) = |x| is continuous but not differentiable at x = 0.
• Find all points of discontinuity and non-differentiability for f(x) = |x - 2| + |x + 2|.

Did You Know?

• The concept of continuity was first formalized by mathematicians like Cauchy and Weierstrass.
• Some functions can be continuous everywhere but differentiable nowhere!

Glossary

• Limit: The value a function approaches as the input approaches a certain point.
• Derivative: The rate at which a function changes at a point.
• Left-hand limit (LHL): The value approached from the left side of a point.
• Right-hand limit (RHL): The value approached from the right side of a point.

Answers to Practice Questions

1. Yes, f(x) = 3x + 2 is continuous at x = 1 because LHL = RHL = f(1) = 5.
2. No, f(x) = |x| is not differentiable at x = 0 because the left and right derivatives are not equal.
3. f'(x) = 3x²; it is continuous and differentiable everywhere, including at x = 0.
4. f(x) = |x| is continuous but not differentiable at x = 0.
5. Every differentiable function is continuous, but every continuous function may not be differentiable.

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Master continuity and differentiability to build a strong foundation in calculus!