Products Preserve Riemann Integrability

Background

Bounded functions, upper and lower Darboux sums, and partition refinement.

A local estimate proves the theorem

Let $a<b$ , and let $f,g:[a,b]\to\mathbb R$ be Riemann integrable. They are bounded, so choose positive constants $M_f,M_g$ with $|f|\leq M_f$ and $|g|\leq M_g$ on $[a,b]$ . Positive bounds are available even when a function is identically zero.

For a bounded real-valued function $u$ on a nonempty subinterval $I$ , its oscillation is

\begin{aligned} \operatorname{osc}_I(u) &=\sup_{t\in I}u(t)-\inf_{t\in I}u(t)\\ &=\sup_{x,y\in I}|u(x)-u(y)|. \end{aligned}

The second equality follows by choosing values arbitrarily close to the supremum and infimum; every difference is bounded by their separation.

For $x,y\in I$ , the identity

\begin{aligned} &f(x)g(x)-f(y)g(y)\\ &\quad=(f(x)-f(y))g(x)\\ &\qquad+f(y)(g(x)-g(y)) \end{aligned}

separates the change in the product into changes in its factors. Taking absolute values, using the bounds, and taking the supremum over $x,y$ gives

\begin{aligned} \operatorname{osc}_I(fg) &\leq M_g\operatorname{osc}_I(f)\\ &\quad+M_f\operatorname{osc}_I(g). \end{aligned}

For a bounded function $u:[a,b]\to\mathbb R$ and a partition $P=\{a=x_0<\cdots<x_n=b\}$ , write $I_i=[x_{i-1},x_i]$ and $\Delta x_i=x_i-x_{i-1}$ . The Darboux gap is

\begin{aligned} D_P(u)&=U(P,u)-L(P,u)\\ &=\sum_{i=1}^{n}\operatorname{osc}_{I_i}(u)\,\Delta x_i. \end{aligned}

Here $U$ and $L$ are the upper and lower Darboux sums. Thus each interval contributes its oscillation multiplied by its length.

Fix $\varepsilon>0$ and set $\rho=\varepsilon/(M_f+M_g+1)$ . Integrability gives partitions $P_f,P_g$ with $D_{P_f}(f)<\rho$ and $D_{P_g}(g)<\rho$ . Their common refinement $P=P_f\cup P_g$ still satisfies $D_P(f)<\rho$ and $D_P(g)<\rho$ : refinement cannot increase a Darboux gap, since each smaller interval has no greater oscillation than the interval containing it.

Multiply the local estimate by each interval's length and sum:

\begin{aligned} D_P(fg)&\leq M_gD_P(f)+M_fD_P(g)\\ &<(M_f+M_g)\rho\\ &<\varepsilon. \end{aligned}

The product is bounded by $M_fM_g$ , and $\varepsilon$ was arbitrary. The Darboux criterion therefore proves that $fg$ is Riemann integrable on $[a,b]$ .

Continue with the full exposition

The full exposition develops the oscillation lemma, includes diagrams, and extends the method to other Lipschitz constructions.

Revision history · September 22, 2026; October 7, 2026

Revised September 22, 2026: the product proof chooses positive bounds for the factors, making its strict estimate valid also when both functions vanish. Nonempty intervals are explicit, and the opening proof outline has been reflowed. The PDF and TeX source agree. Revised October 7, 2026: condensed the explanatory prose and made the hypotheses of the concluding Lipschitz principle explicit.

More analysis & topology All expositions

Background

A local estimate proves the theorem

Continue with the full exposition

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