# Improper or Undefined

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

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