Let $[a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
| (i) | $a < b$ |
| (ii) | $f : [a, b] \to \mathbb{R}$ is a D1760: Riemann integrable real function |
| (iii) | \begin{equation} F : [a, b] \to \mathbb{R}, \quad F(x) = \int^x_a f(t) \, d t \end{equation} |
Then
| (1) | $F$ is a D5231: Standard-continuous real function |
| (2) | If $f$ is a D5231: Standard-continuous real function at $x_0 \in [a, b]$, then $F'(x_0) \neq \emptyset$ and $F'(x_0) = f(x_0)$ |