Algebra of Riemann integrable functions

Let $f,g:[a,b]\to\mathbb{R}$ be Riemann integrable .

Proposition:

• For any $\alpha,\beta\in\mathbb{R}$, the function $\alpha f+\beta g$ is Riemann integrable.
• The product $fg$ is Riemann integrable.
• Consequently, $f^2$ is Riemann integrable.

These closure properties are essential: they guarantee the Riemann integral behaves well under the usual algebraic operations on functions.

Proof sketch: Linearity is standard from linearity of sums and the integrability definition via upper /lower sums . For products, note that Riemann integrable functions are bounded , so $|f|\le M$ and $|g|\le N$. Use the identity $fg=\frac{1}{4}\bigl((f+g)^2-(f-g)^2\bigr)$ to reduce product integrability to integrability of squares, and show that if $h$ is integrable then $h^2$ is integrable (e.g., by continuity of $x\mapsto x^2$ and the Lebesgue criterion , or by oscillation estimates).