Step Function

A function on the Reals $\Bbb{R}$ is a step function if it can be written as a finite linear combination of semi-open intervals $[a,b)\subseteq\Bbb{R}$. Therefore, a step function $f$ can be written as

$$f(x)=\alpha_1 f_1(x)+\cdots+\alpha_n f_n(x),$$

where $\alpha_i\in\Bbb{R}$, $f_i(x)=1$ if $x\in [a_i, b_i)$ and 0 otherwise, for $i=1$, ..., $n$.

See also Heaviside Step Function