Mean Value Theorem for Integrals

Let $I$ be an interval in the sense of Interval in the Real Line, let $[a,b]\subseteq I$ with $a<b$, and let $f:I\to\mathbb{R}$ be continuous on $[a,b]$ in the sense of Continuity on a Closed Interval. 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 Riemann Integrability on a Closed Interval; existence is guaranteed by Continuous Functions on a Closed Interval are Riemann Integrable.