Sum of Integrals on Adjacent Intervals for Continuous Functions

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

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


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


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.



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.