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:

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


Proof

By Continuous Function is Riemann 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:

\(\displaystyle \int_a^b f \left({t}\right) \ \mathrm dt\) \(=\) \(\displaystyle F\left({b}\right) - F\left({a}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle F\left({b}\right) - F\left({c}\right) + F\left({c}\right) - F\left({a}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \int_c^b f \left({t}\right) \ \mathrm dt + \int_a^c f \left({t}\right) \ \mathrm dt\)

... but such a proof would be circular.


Sources