Math 2.0

Real functions Definition

Step function

Builds on Partition of an interval

Definition

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be a function. The function ff is a step function if it is constant on each open subinterval of some partition (x0,…,xn)(x_0, \dots, x_n) of [a,b][a, b]. That is, there are real numbers c1,…,cnc_1, \dots, c_n with f(x)=cif(x) = c_i for all x∈(xi−1,xi)x \in (x_{i-1}, x_i).

Remarks

The values of ff at the points xix_i are arbitrary. A step function is Riemann integrable, and ∫abf=∑i=1nci(xi−xi−1)\int_a^b f = \sum_{i=1}^{n} c_i (x_i - x_{i-1}). The floor function ⌊x⌋\lfloor x \rfloor restricted to [a,b][a, b] is a step function.

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Partition of an interval, definition, no evidence levelPartition of an inter…definitionStep function, definition, no evidence levelStep functiondefinition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Step_function, Definition and first consequences
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