Sum of Complex Integrals on Adjacent Intervals

Theorem
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \C$ be a continuous complex function.

Let $c \in \left[{a \,.\,.\, b}\right]$.

Then:


 * $\displaystyle \int_a^c f \left({t}\right) \ \mathrm dt + \int_c^b f \left({t}\right) \ \mathrm dt = \int_a^b f \left({t}\right) \ \mathrm dt$

Proof
From Continuous Complex Function is Complex Riemann Integrable, it follows that all three complex integrals are well defined.

From Real and Imaginary Part Projections are Continuous, it follows that $\operatorname{Re}: \C \to \R$ and $\operatorname{Im}: \C \to \R$ are continuous functions.

From Continuity of Composite Mapping, it follows that $\operatorname{Re} \circ f: \R \to \R$ and $\operatorname{Im} \circ f: \R \to \R$ are continuous real functions.

Then: