The Euler integral of a function 
that is piecewise-constant with finitely many discontinuities
is the sum of
![f(x)-1/2[f(x_+)+f(x_-)]](/images/equations/EulerIntegral/NumberedEquation1.svg) |
over the finitely many discontinuities of 
. Euler integration was defined by Schanuel and subsequently
explored by Rota, Chen, and Klain. The 
-dimensional Euler integral can be defined for classes of functions
. Euler integration is additive,
so the Euler integral of 
equals the sum of the Euler integrals of 
and 
.
