Bounded interval/Step function/Step integral/Definition

Step integral

Let ${}I$ be a real interval with endpoints ${}a,b\in \mathbb {R}$, and let

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

denote a step function for the partition ${}a=a_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<a_{n}=b$, with the values ${}t_{i}$, ${}i=1,\ldots ,n$. Then

${}T:=\sum _{i=1}^{n}t_{i}(a_{i}-a_{i-1})\,$

is called the step integral of ${}t$ on ${}I$.