# 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

- 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.): $\S 4.3$