Theorems on Integrability

Theorem 1

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.

Proof Hint

• Isolate a partition on the required end.
• Choose $x_1$ or $x_{n-1}$ such that $\Delta{x}<\frac{\epsilon}{4M}$ where $M$ is an upper or lower bound.

Theorem 2

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.