CBSE Class 12 Mathematics Continuity and Differentiability Q #2096
COMPETENCY BASED
EVALUATE
1 Marks ASSERTION REASON
Assertion: If a function is continuous on a closed interval, then it must be differentiable on that interval.
Reason: Continuity ensures smoothness, which guarantees differentiability.
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is NOT the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.
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Detailed Solution

Step 1: Analyze the Assertion

The assertion states that if a function is continuous on a closed interval, it must be differentiable on that interval. We must test this against known counterexamples in calculus.

Step 2: Evaluate the Counterexample

Consider the function f(x) = |x| on the interval [-1, 1]. This function is continuous everywhere on the interval, including at x = 0. However, the derivative at x = 0 does not exist because the left-hand derivative is -1 and the right-hand derivative is 1. Thus, continuity does not imply differentiability.

Step 3: Evaluate the Reason

The reason claims that continuity ensures smoothness, which guarantees differentiability. This is mathematically incorrect. Continuity is a necessary condition for differentiability, but it is not a sufficient condition. Smoothness is a property associated with the existence of derivatives, not a cause of them.

Step 4: Conclusion

Since the assertion is false and the reasoning provided is also false, the statement is incorrect.

Final Answer: Both Assertion and Reason are False.

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Pedagogical Audit
Bloom's Analysis: This is an EVALUATE question because it requires the student to critique the validity of a mathematical statement by identifying counterexamples and logical fallacies.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the relationship between continuity and differentiability, which is a fundamental concept in Calculus.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It moves beyond rote memorization of theorems to test the student's ability to distinguish between necessary and sufficient conditions.

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