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NCERT Solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability

NCERT Solutions for Class 12 Mathematics Chapter 5 Continuity and Differentiability covers the concepts of continuity and differentiability of a function, rules of differentiation including the chain rule and logarithmic differentiation, and key theorems like Rolle's Theorem and the Mean Value Theorem.

Overview of Topics & Subtopics (Click to Jump)

1. Continuity of a Function

Short Note:

• A function f(x) is continuous at a point x = a if the left-hand limit, right-hand limit, and the value of the function at that point are all equal.

• Algebraic combinations (sum, difference, product, quotient) of continuous functions are also continuous wherever they are defined.

2. Differentiability and Rules of Differentiation

Short Note:

• A function differentiable at a point is necessarily continuous there, though the converse need not be true, as seen in functions like |x| at x = 0.

• Important techniques include the chain rule for composite functions, implicit differentiation for equations not explicitly solved for y, and logarithmic differentiation for expressions of the form [f(x)]^g(x).

3. Rolle's Theorem and Mean Value Theorem

Short Note:

• Rolle's Theorem states that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) where f'(c) = 0.

• The Mean Value Theorem generalizes this, stating that there exists a point c where f'(c) equals the average rate of change of f over [a, b].

NCERT Textbook Exercise Questions

What is the difference between continuity and differentiability of a function at a point?
Answer: A function is continuous at a point if there is no break, jump, or hole in its graph at that point, meaning the limit of the function equals its actual value there. Differentiability, on the other hand, requires the existence of a well-defined tangent (or derivative) at that point, which is a stronger condition; every differentiable function is continuous, but not every continuous function is differentiable, as seen with sharp corners like in |x|.
State Rolle's Theorem and explain its geometrical meaning.
Answer: Rolle's Theorem states that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0. Geometrically, this means that somewhere between two points on the curve with equal heights, there must be at least one point where the tangent to the curve is horizontal.
When is logarithmic differentiation used, and why?
Answer: Logarithmic differentiation is used for functions of the form y = [f(x)]^g(x), where both the base and the exponent are functions of x, making direct differentiation using standard rules difficult. By taking the natural logarithm of both sides first, the expression is converted into a product, ln y = g(x) · ln f(x), which can then be differentiated more easily using the product rule and chain rule.
Verify that the function f(x) = |x| is continuous but not differentiable at x = 0.
Answer: The function f(x) = |x| is continuous at x = 0 since the left-hand limit, right-hand limit, and the value f(0) = 0 are all equal. However, its left-hand derivative at x = 0 is -1 while its right-hand derivative is +1; since these two values are not equal, the function does not have a well-defined derivative at x = 0, making it non-differentiable there despite being continuous.
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Class 12 › Mathematics › Continuity and Differentiability
Chapter notesRead the chapterPractice MCQsCheck what you rememberChapter test10 questions on one pageOnline testTest 1, 2 and 3, one question at a timePYQPrevious year questionsStudy materialPDF notes
Topics in this chapter: Continuity of a FunctionDifferentiability and Rules of DifferentiationRolle's Theorem and Mean Value Theorem