Definite Integral on Zero Interval

Theorem
Let $f$ be a real function which is defined on the closed interval $\left[{a \,.\,.\, b}\right]$.

Then:
 * $\displaystyle \forall c \in \left[{a \,.\,.\, b}\right]: \int_c^c f \left({t}\right)\ \mathrm dt = 0$

Proof
Follows directly from the definition of definite integral.

There is only one subdivision of $\left[{c \,.\,.\, c}\right]$ and that is $\left\{{c}\right\}$.

Both the lower sum and upper sum of $f \left({x}\right)$ on $\left[{c \,.\,.\, c}\right]$ belonging to the subdivision $\left\{{c}\right\}$ are equal to zero.

Hence the result.