Question:
For a Riemann integrability, the condition of continuity is
For a Riemann integrability, the condition of continuity is
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Continuity $\implies$ Integrability (Sufficient), but Integrability $\not\implies$ Continuity (Not Necessary).
Updated On: Jul 3, 2026
- necessary
- sufficient
- not necessary
- Insufficient
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The Correct Option is B
Solution and Explanation
Step 1: Concept
A function does not have to be continuous to be Riemann integrable, but every continuous function is integrable.
Step 2: Meaning
A "sufficient" condition means that if the condition (continuity) is met, the result (integrability) is guaranteed.
Step 3: Analysis
Functions with a finite number of jump discontinuities are also integrable, meaning continuity is not "necessary". However, continuity on a closed interval $[a,b]$ always implies integrability.
Step 4: Conclusion
Thus, continuity is a sufficient condition for Riemann integrability. Final Answer: (B)
A function does not have to be continuous to be Riemann integrable, but every continuous function is integrable.
Step 2: Meaning
A "sufficient" condition means that if the condition (continuity) is met, the result (integrability) is guaranteed.
Step 3: Analysis
Functions with a finite number of jump discontinuities are also integrable, meaning continuity is not "necessary". However, continuity on a closed interval $[a,b]$ always implies integrability.
Step 4: Conclusion
Thus, continuity is a sufficient condition for Riemann integrability. Final Answer: (B)
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