Prove that if $f$ is a bounded function on $[a,b]$, and for each $c \in (a,b)$, $f \in R[a,c]$, then $f \in R[a,b]$.
Prove for bounded $f$, if $f:[a,r] \to \mathbb{R}$ is Riemann-Integrable for $r \in [a,b)$ then $f:[a.b] \to \mathbb{R}$ is Riemann-Integrable