Interval/Real function/Step function/Definition

Step function

Let ${}I$ be a real interval with endpoints ${}a,b\in \mathbb {R}$. Then a function

$t\colon I\longrightarrow \mathbb {R}$

is called a step function, if there exists a partition

${}a=a_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<a_{n}=b\,$

of ${}I$ such that ${}t$ is constant on every open interval ${}]a_{i-1},a_{i}[$.