Fubini’s Theorem (Riemann, continuous case): Let $f:[a,b]\times[c,d]\to\mathbb{R}$ be continuous . Define $g(x)=\int_c^d f(x,y)\,dy \quad (x\in[a,b]), \qquad h(y)=\int_a^b f(x,y)\,dx \quad (y\in[c,d]).$ Then $g$ and $h$ are continuous, and the double integral exists and satisfies $ \int_a^b\left(\int_c^d f(x,y),dy\right)dx

\int_{[a,b]\times[c,d]} f(x,y),d(x,y)

\int_c^d\left(\int_a^b f(x,y),dx\right)dy. $

This theorem justifies computing multiple integrals by integrating one variable at a time via iterated integrals , a core technique in analysis and applications.

Proof sketch: Continuity on the compact rectangle implies uniform continuity and boundedness. Approximate $f$ uniformly by step functions on rectangles (or compare upper /lower sums ). For step functions the statement is immediate by finite additivity. Uniform approximation plus the uniform convergence-and-integration principle yields the equality for $f$.