Sum of Integrals on Adjacent Intervals for Continuous Functions

Theorem

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

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

Then:

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

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:

\(\ds \int_a^b \map f t \rd t\)

\(=\)

\(\ds \map F b - \map F a\)

\(\ds \)

\(=\)

\(\ds \map F b - \map F c + \map F c - \map F a\)

\(\ds \)

\(=\)

\(\ds \int_c^b \map f t \rd t + \int_a^c \map f t \rd t\)

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.): $\S 4.3$