The Riemann integral allows only the integration of functions over bounded real intervals, i.e for closed real intervals of the type $[a,b]$, $a\le b$ and for unbounded functions on semi-open or open intervals $[a,b)$, $(a,b]$, and $(a,b).$ The concept of the Riemann integral fails to work so far for the following cases:
The following definition clarifies, under which certain circumstances the concept of Riemann integral can be extended to even these special cases. This extended concept of the Riemann integral is known as improper intergral.