Additivity of the Riemann Integral on Adjacent Intervals

Let $a,b,c\in\mathbb{R}$ satisfy $a\le b\le c$, and let $f:[a,c]\to\mathbb{R}$ be Riemann integrable on $[a,c]$. Then the restrictions $f|_{[a,b]}:[a,b]\to\mathbb{R}$ and $f|_{[b,c]}:[b,c]\to\mathbb{R}$ are Riemann integrable on $[a,b]$ and $[b,c]$, respectively, and

In particular,