Let $I$ be an interval in the sense of \ref{def:interval-real-line-c54-2026a}, let $[a,b]\subseteq I$ with $a<b$, and let $f:I\to\mathbb{R}$ be continuous on $[a,b]$ in the sense of \ref{def:continuity-closed-interval-c54-2026a}. Then there exists $c\in[a,b]$ such that $\int_a^b f(x)\,dx=f(c)(b-a),$ where the integral is understood as in \ref{def:riemann-integrable-closed-interval-c54-2026a}; existence is guaranteed by \ref{lem:continuous-implies-riemann-integrable-c54-2026a}.