# Proving function is not integrable

• Nov 19th 2011, 08:31 PM
BrownianMan
Proving function is not integrable
S(x) = x - n where n<= x <= n+1

How do I show that there is no function F such that F'=S?
• Nov 19th 2011, 10:49 PM
chisigma
Re: Proving function is not integrable
Quote:

Originally Posted by BrownianMan
S(x) = x - n where n<= x <= n+1

How do I show that there is no function F such that F'=S?

S(x) is known as 'fractional part of x'...

Fractional Part -- from Wolfram MathWorld

... and it seems not to exist a valid reason for which it is not integrable...

Kind regards

$\chi$ $\sigma$
• Nov 20th 2011, 03:08 PM
Jose27
Re: Proving function is not integrable
Quote:

Originally Posted by BrownianMan
S(x) = x - n where n<= x <= n+1

How do I show that there is no function F such that F'=S?

$S$ can't be a derivative because it has jump discontinuities at $\mathbb{Z}$, and derivatives satisfy the mean value property. On the other hand it's integrable over any bounded interval. I suggest youu review what the fundamental theorem of calculus actually says.