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 D1760: Riemann integrable real function |
(iii) | $a = c_0, c_1, \dots, c_N = b$ is an D3687: Implicit interval partition of $[a, b]$ |
Then
(1) | $[c_{n - 1}, c_n] \to \mathbb{R}, \quad x \mapsto f(x)$ is a D1760: Riemann integrable real function for each $n \in 1, \ldots, N$ |
(2) | \begin{equation} \int^b_a f(x) \, d x = \sum_{n = 1}^N \int^{c_n}_{c_{n - 1}} f(x) \, d x \end{equation} |