Sum of Integrals on Adjacent Intervals for Continuous Functions

Theorem
Let $f$ be a real function which is continuous on any closed interval $I$.

Let $a, b, c \in I$.

Then:
 * $\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$

Proof
By Continuous Real Function is Darboux Integrable, $f$ is integrable on $I$.

The result follows by application of Sum of Integrals on Adjacent Intervals for Integrable Functions.

Comment
This proof would be very simple if we were to use the Fundamental Theorem of Calculus:

but such a proof would be circular.