Sum of Integrals on Adjacent Intervals for Continuous Functions

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

Let $$a < c < b$$.

Then $$\int_a^c f \left({t}\right) dt + \int_c^b f \left({t}\right) dt = \int_a^b f \left({t}\right) dt$$.

Proof
Let $$P_1$$ and $$P_2$$ be any subdivisions of $$\left[{a \,. \, . \, c}\right]$$ and $$\left[{c \,. \, . \, b}\right]$$ respectively.

Then $$P_1 \cup P_2$$ is a subdivision of $$\left[{a \,. \, . \, b}\right]$$.

Let $$L \left({P}\right)$$ be the lower sum of a given subdivision $$P$$, and $$U \left({P}\right)$$ its upper sum.

It is clear from the definitions of upper sum and lower sum that:
 * $$L \left({P_1}\right) + L \left({P_2}\right) = L \left({P_1 \cup P_2}\right)$$;
 * $$U \left({P_1}\right) + U \left({P_2}\right) = U \left({P_1 \cup P_2}\right)$$.

It follows from the definition of definite integral that:
 * $$L \left({P_1}\right) \le \int_a^c f \left({t}\right) dt \le U \left({P_1}\right)$$;
 * $$L \left({P_2}\right) \le \int_c^b f \left({t}\right) dt \le U \left({P_2}\right)$$.

The result follows.