# 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.

$\blacksquare$

## Comment

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

 $\ds \int_a^b \map f t \rd t$ $=$ $\ds \map F b - \map F a$ $\ds$ $=$ $\ds \map F b - \map F c + \map F c - \map F a$ $\ds$ $=$ $\ds \int_c^b \map f t \rd t + \int_a^c \map f t \rd t$

but such a proof would be circular.