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?
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?
Option 1 -
f is differentiable at every point in [0,∞)
Option 2 -
f is continuous everywhere except at the integer points in [0,∞).
Option 3 -
f is continuous at every point in [0,∞) and differentiable except at the integer points.
Option 4 -
f is both continuous and differentiable except at the integer points in [0,∞).
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1 Answer
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Correct Option - 2
Detailed Solution: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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