In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 on the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s is called a step function (or staircase function) if it can be written as a finite linear combination

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

 of indicator functions of interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

s. Informally speaking, a step function is a piecewise

Piecewise

On mathematics, a piecewise-defined function is a function whose definition changes depending on the value of the independent variable...

 constant function

Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...

 having only finitely many pieces.
A function is called a step function if it can be written as
for all real numbers

where are real numbers, are intervals, and is the indicator function of :


In this definition, the intervals can be assumed to have the following two properties:

1. The intervals are disjoint, for
2. The union
   Union (set theory)
   In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
   
    of the intervals is the entire real line,

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold.