= Step function
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On a real interval, a step function is constant on the open cells of a finite <partition of an interval>, with endpoint values specified separately. Equivalently, up to those finitely many point values it is a finite linear combination of interval <indicator functions>. For example, a function equal to two on $[0,1/4)$, one on $[1/4,3/4)$ and zero on $[3/4,1]$ is a step function. Its integral is the sum of rectangle heights times widths.
A <simple function> is more general in measure theory: its finitely many value sets can be arbitrary <measurable sets>. The <Dirichlet function> $\mathbf1_{\mathbb Q}$ on $[0,1]$ is simple but is not an interval step function, because every nonempty interval contains both rational and irrational points. Merely being constant on a finite or countable measurable partition therefore does not characterize the usual interval-based step functions.
Countably many jumps can be considered under explicitly stated extensions of the term. The finite-partition convention is the one used for elementary Riemann approximations; the measurable finite-range convention belongs to <simple functions> in Lebesgue integration.
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