Let $ [a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
| (i) | \begin{equation} a < b \end{equation} |
| (ii) | $f : [a, b] \to \mathbb{R}$ is a D5231: Standard-continuous real function |
Then
| (1) | \begin{equation} \exists \, c \in [a, b] : \int_a^b f(x) \, d x = f(c)(b - a) \end{equation} |
| (2) | \begin{equation} \exists \, c \in [a, b] : \frac{1}{b - a} \int_a^b f(x) \, d x = f(c) \end{equation} |