"Suppose $h(x)$ is a function that measures the local mesh size near the point $x$. In particular, we will assume that $h$ is a piecewise linear function satisfying
\begin{equation}h(x_j) = h_j + h_{j+1}\end{equation}
where $h_j = x_j - x_{j-1}$ (and we set $h_{n+1}=h_n$ and $h_0=h_1$). Note that for all $j = 1, \ldots, n$
\begin{equation}h(x)\geq h_j,\quad\forall x \in [x_{j-1}, x_j],\end{equation}
since this holds at each endpoint of the interval and $h$ is linear between them."
From Chapter 0, Brenner & Scott