Sum of Integrals on Adjacent Intervals for Continuous Functions

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Theorem

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

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


Then:

$\displaystyle \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$


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:

\(\displaystyle \int_a^b \map f t \rd t\) \(=\) \(\displaystyle \map F b - \map F a\)
\(\displaystyle \) \(=\) \(\displaystyle \map F b - \map F c + \map F c - \map F a\)
\(\displaystyle \) \(=\) \(\displaystyle \int_c^b \map f t \rd t + \int_a^c \map f t \rd t\)

... but such a proof would be circular.


Sources