Let \(I=\left[a,b\right]\subset \mathbb{R}\) be a closed interval. A partition of \(I\) is a finite set of bounded intervals contained in \(I\) such that every \(x \in I\) lies in exactly one of the intervals. The intervals may be open, closed or half-open, and may be single points \([x,x]=\{x\}.\) For example, \(\{[0,\tfrac{1}{2}),[\tfrac{1}{2},1]\}\) is a partition of \([0,1]\) whereas \(\{[0,\tfrac{1}{2}],[\tfrac{1}{2},1]\}\) is not, because \(\tfrac{1}{2}\) lies in both intervals.