Theorems on Integrability

Theorem

Suppose $f:[a,b]\rightarrow\mathbb{R}$ is bounded, and integrable on $[c,b]$ for all $c\in(a,b)$. Then $f$ is integrable on $[a,b]$. Also valid for the other end.

Theorem

Suppose $f:[a,b]\rightarrow\mathbb{R}$ is bounded, and continuous on $[c,b]$ for all $c\in(a,b)$. Then $f$ is integrable on $[a,b]$. Also valid for the other end.