Definition

Let $f:[0,1]\to \mathbb{R}$ and let $\pi=(t_{0},\dots,t_{N})$ be a subdivision of $[0,1]$. For each $i\in\{0,\dots,N-1\}$ let $$\begin{align*} m_{i}&=\inf\{f(t):t\in[t_{i},t_{i+1}]\}\\ M_{i}&=\sup\{f(t):t\in[t_{i},t_{i+1}]\} \end{align*}$$We define the Lower Darboux sum of $f$ with respect to $\pi$ by $$S_{f,\pi}^{\mathscr{l}}=\sum_{i=1}^{N-1}m_{i}(t_{i+1}-t_{i})$$ and the Upper Darboux sum of $f$ with respect to $\pi$ by $$S_{f,\pi}^{\mathscr{u}}=\sum_{i=1}^{N-1}M_{i}(t_{i+1}-t_{i})$$