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Some Properties of Definite Integrals

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Some Properties of Definite Integrals Class 12th

Let f: [0,∞) → [0,∞) be defined as f(x) = ∫? [y]dy where [x] is the greatest integer less than or equal to x. Which of the following is true?
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Raj Pandey

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f (x) = ∫? [y]dy. For x∈ [n, n+1), [y]= [x]=n.
f (x) = Σ (k=0 to n-1) ∫? ¹ k dy + ∫? n dy = Σk + n (x-n).
f (x) is continuous at integers. f' (x)=n= [x]. Not differentiable at integers.

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