Mathematical Analysis Fundamentals

(9.10)

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 $\mathcal{P}=\{x_0\text{,}\dots \text{,}{x}_n\}$