Splitting up an interval in subintervals and defining functions that are constant on them.

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Concept |
Content |

Real Numbers |
In a real analysis, the real numbers are the largest number set we need. They satisfy axioms that represent the idea of a number line. |

Maps |
Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed for the domain and the codomain. |

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**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 \(\mathcal{T}([a,b])\).

It can be readily verified that for two step functions \(f_1,f_2\in \mathcal{T}([a,b])\) holds \(f_1+f_2\in \mathcal{T}([a,b])\). As well,
we have \(\lambda f_1\in
\mathcal{T}([a,b])\). Hence, \(\mathcal{T}([a,b])\) is a vector space.
Furthermore, since step functions only attain finitely many values, they
are bounded.

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