Differential Calculus> continuity diff!!!...

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?

ANUJ VISHAL , 14 Years ago

Grade 12

1 Answers

Jitender Singh

Last Activity: 10 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

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Differential Calculus> continuity diff!!!...

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?

ANUJ VISHAL , 14 Years ago

Jitender Singh

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