This function has the following properties.

Theorem 9.25

*If* $f\in R(a\text{,}b)$, *then* $F\in C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$, *where* $F$ *is defined by Eq.* *(9.10)* *.*

Proof

Let $a\le x<y\le b$. Then

$\mid F(y)-F(x)\mid =\left|{\int}_{x}^{y}f(t)\mathit{dt}\right|\le {\int}_{x}^{y}\mid f(t)\mid \mathit{dt}\le \parallel f{\parallel}_{B}(y-x)\text{,}$

proving that $F\in C(a\text{,}b)$. If $\mathcal{P}=\{{x}_{0}\text{,}\dots \text{,}{x}_{n}\}$