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Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral

$\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined?

If we take it as a legitimate function for improper Riemann integral ,then as a limit this

seems to be true.Otherwise it is undefined.Do we allow infinite values for improper Riemann

integral? Thanks in advance..

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You may be interested in looking up Dirac measure. en.wikipedia.org/wiki/Dirac_measure –  Dylan Yott Feb 20 '13 at 22:29
1  
Usually, one accepts only real-valued functions for proper Riemann integrals, that is your $f$ is defined only on $[0,1)$. –  Hagen von Eitzen Feb 20 '13 at 22:29
    
@HalilDuru Consider my answer wrong, I didn't notice you were asking about the Riemann integral, sorry –  AndreasT Feb 20 '13 at 22:38
    
ok, no problem.. –  Halil Duru Feb 20 '13 at 22:38

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