Let $a,b\in\mathbb{R}$ with $a<b$. A partition of $[a,b]$ is a finite sequence $P=(x_0,x_1,\dots,x_n)$ of real numbers such that $a=x_0<x_1<\cdots<x_n=b$. The mesh of $P$ is defined by $|P|=\max_{1\le i\le n}(x_i-x_{i-1}).$