Let $[a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
| (i) | \begin{equation} a \leq b \end{equation} |
| (ii) | $f : [a, b] \to \mathbb{R}$ is a D1760: Riemann integrable real function |
| (iii) | $c \in [a, b]$ |
Then
| (1) | $[a, c] \to \mathbb{R}, \quad x \mapsto f(x)$ is a D1760: Riemann integrable real function |
| (2) | $[c, b] \to \mathbb{R}, \quad x \mapsto f(x)$ is a D1760: Riemann integrable real function |
| (3) | \begin{equation} \int^b_a f(x) \, d x = \int^b_c f(x) \, d x + \int^c_a f(x) \, d x \end{equation} |