Let
be a compact interval, and let
-
be a function. Then the following statements are equivalent.
- The function
is
Riemann-integrable.
- There exists a partition
,
such that the restrictions
are Riemann-integrable.
- For every partition
,
the restrictions
are Riemann-integrable.
In this situation, the equation
-
![{\displaystyle {}\int _{a}^{b}f(t)\,dt=\sum _{i=1}^{n}\int _{a_{i-1}}^{a_{i}}f_{i}(t)\,dt\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb23748503e1936daf27d5119552be987506a0d0)
holds.