Positive Part of Darboux Integrable Function is Integrable/Negative Part

Theorem

Let $f$ be a real function that is Darboux integrable over $\closedint a b$.

Let $f^-$ be the negative part of $f$.

Then $f^-$ is Darboux integrable over $\closedint a b$.

Proof

$f^-$ is the positive part of $\map g x = -\map f x$.

From Linear Combination of Definite Integrals, it follows that:

$\ds \int_a^b \map g x \rd x = -\int_a^b \map f x \rd x$

Therefore, by Positive Part of Darboux Integrable Function is Integrable, $f^-$ is Darboux integrable over $\closedint a b$.

$\blacksquare$