Differentiability
Assertion: every differentiable function is continuous. Reason: differentiability requires the difference quotient limit to exist; which option correctly judges these?
Select the correct option:
Solution
Both true and reason correctly explains assertion
This assertion-reason item tests the logical chain linking differentiability to continuity, a cornerstone theorem of single-variable calculus. The assertion is true: if f is differentiable at a, then \lim_{x \to a} (f(x) - f(a)) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}(x-a) = f'(a) \cdot 0 = 0, forcing f(x) \to f(a), which is exactly continuity. The reason is also true, since differentiability is defined precisely by the existence of that difference-quotient limit. Crucially, the reason supplies the very mechanism that proves the assertion, because the existence of the finite quotient is what lets the product with (x-a) collapse to zero. Hence the reason correctly explains the assertion. The second option fails because the explanation is genuine, not incidental. The third is wrong as the assertion holds. The fourth is wrong as the assertion is not false. The governing pattern is the differentiability-implies-continuity theorem. Plausibility check: the converse fails, since |x| is continuous yet not differentiable at 0, which is consistent with continuity being the weaker, implied property rather than the stronger one.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- differentiability
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
Both true and reason correctly explains assertion
This assertion-reason item tests the logical chain linking differentiability to continuity, a cornerstone theorem of single-variable calculus. The assertion is true: if f is differentiable at a, then \lim_{x \to a} (f(x) - f(a)) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}(x-a) = f'(a) \cdot 0 = 0, forcing f(x) \to f(a), which is exactly continuity. The reason is also true, since differentiability is defined precisely by the existence of that difference-quotient limit. Crucially, the reason supplies the very mechanism that proves the assertion, because the existence of the finite quotient is what lets the product with (x-a) collapse to zero. Hence the reason correctly explains the assertion. The second option fails because the explanation is genuine, not incidental. The third is wrong as the assertion holds. The fourth is wrong as the assertion is not false. The governing pattern is the differentiability-implies-continuity theorem. Plausibility check: the converse fails, since |x| is continuous yet not differentiable at 0, which is consistent with continuity being the weaker, implied property rather than the stronger one.
This easy difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of differentiability. It appeared in the 2025 exam.
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