# 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

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.8$

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 4.3$