If some [[Function]] with [[Codomain]] in the [[Real Numbers|Real Numbers]], $f : [a,b] \to \R$ is [[Continuous]] on the [[Closed Interval]] $[a, b]$ and is [[Derivative|Differentiable]] on the [[Open Interval]] $(a, b)$, then the [[Average|Mean]] Value states that there [[Existential Quantifier|exists]] a number in the [[Closed Interval]] $C \in [a,b]$ such that $f(C)$ is the average value of $f$ in the [[Closed Interval]] $[a,b]$. $\huge f(C) \cdot (b-a) = \int_{a}^b f(x)\d{x}\iff f \text{ is continuous} $