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F(x)=integrate [f(t)]dt from 0 to x where x>0 and [.] denotes greatest int. func. .Then, is f(x) continuous but not diff. for x=1,2,3,........ If yes,how?

F(x)=integrate [f(t)]dt from 0 to x where x>0 and [.] denotes greatest int. func. .Then,


is f(x) continuous but not diff. for x=1,2,3,........ If yes,how?

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 years ago
Ans:
f(t) should be given. f(t) = t
F(x) = \int_{0}^{x}[t]dt
Apply the newton leibniz’s theorm here, we have,
F'(x) = [x].1 - [0].0 = [x]
F'(x) = f(x)
f(x) = [x]
To check the continuity at integer ‘a’
LHL = RHL
\lim_{x\rightarrow a^{-}}[x] = \lim_{x\rightarrow a^{+}}[x]
LHL = \lim_{x\rightarrow a^{-}}[x] = a-1
RHL = \lim_{x\rightarrow a^{+}}[x] = a
So f(x) is not continuous at integer ‘a’. So it will definitely will not be differentiable at integer ‘a’.
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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