Partitions and Step Functions

Definition 1. Let \([a,b]\subset\mathbb{R}\). A set \(\{x_{0},x_{1},\ldots,x_{n}\}\) is called a decomposition or partition of \([a,b]\) if \[a=x_0<x_1<x_2<\ldots<x_{n-1}<x_n=b.\]

Definition 2 (Step function). \(f:[a,b]\to\mathbb{R}\) is called a step function if it is piecewisely constant, i.e. there exists a decomposition \(\{x_{0},\ldots,x_{n}\}\) of \([a,b]\) and some \(c_1,\ldots,c_n\in\mathbb{R}\) such that for all \(i=1,\ldots,n\) holds \[f(x)=c_i\text{ for all }x\in(x_{i-1},x_i).\] The set of step functions on \([a,b]\) is denoted by \(\mathcal{T}([a,b])\).