Step function

A function that is constant on each subinterval of some partition.

Step function

A step function on $[a,b]$ is a function $s:[a,b]\to\mathbb R$ for which there exists a partition $P=\{x_0,\dots,x_n\}$ and constants $c_1,\dots,c_n$ such that

s(x)=c_i \quad \text{for all } x\in(x_{i-1},x_i),\ i=1,\dots,n.

The values of $s$ at the partition points $x_i$ can be chosen arbitrarily without changing this property.

Step functions are basic building blocks in the theory of the Riemann integral ; in particular, they are Riemann integrable and play a role analogous to a simple function in measure theory.

Examples:

Any constant function $s(x)=c$ on $[a,b]$ is a step function (take $P=\{a,b\}$ ).
On $[-1,1]$ , the function $s(x)=0$ for $x<0$ and $s(x)=1$ for $x>0$ is a step function (take $P=\{-1,0,1\}$ ).