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 , one on and zero on 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 on 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.
A continuous function on a compact interval is uniformly continuous by the Heine-Cantor theorem. Choose a partition of an interval whose widths lie below the input tolerance for a desired output error . On each half-open partition subinterval, use the value of the function at its left endpoint, and assign the rightmost endpoint its own value. This defines a step function whose error is strictly below at every input. Finitely assigned partition-point values are allowed and do not alter the Riemann integral.
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A step function is a type of piecewise function that changes its value at specific intervals, resulting in a graph that looks like a series of steps. These intervals can be defined by any rules, leading to a function that stays constant over each interval before jumping to a new value at the boundaries. ### Key Characteristics of Step Functions: 1. **Piecewise Definition**: A step function can be defined using different constant values over different ranges of the input variable.
