Step function
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In mathematics, a function on the reals R is a step function if it can be written as a finite linear combination of characteristic functions of semi-open intervals.
Let the following quantities be given:
- a sequence of coefficients
- <math>\{\alpha_0, \dots, \alpha_n\}\subset \mathbb{R},\; n \in \mathbb{N} \setminus \{0\}<math>
- a sequence of interval margins
- <math>\{x_1 < \dots < x_{n-1}\} \subset \mathbb{R}<math>
- a set of functions <math>p_i:\mathbb{R}\to\mathbb{R}, \;\; i \in \overline{0,n}<math> such that:
- <math> p_0(x) = \left\{
- <math> \forall i \in \overline{1, n-1} \;\;
- <math> p_n(x) = \left\{
Definition: Given the notations above, a function f:R→R is a step function if and only if it could be written as
- <math>
f(x) = \sum\limits_{i=0}^n \alpha_i \cdot p_i(x). <math>
Note: Step functions so defined are piecewise constant.