Step function

A function of an interval is a step function (or simple mistake or elementary function ), if there are disjoint intervals, so that

And on the interval is constant.

Step functions are also used for the approximation of integrals. The integral of a step function is

Defined. The advantage here is that you can do without limit process and has only finite sums. In the formula denotes the length of the interval, that is for example,. With we denote the value of on the interval.

Already by the simple definition of the integral of a step function has gained a strong mathematical tool: Every bounded, continuous function, can be approximated arbitrarily closely by a step function. So the integral of this function can be approximated with arbitrary precision. This fact is an important foundation for the definition of the Riemann integral. In this way, Jean Gaston Darboux has simplified the introduction of the Riemann integral.

Examples

  • The Heaviside function is 0 for any negative number, otherwise 1
  • Rectangular function
  • Floor function and ceiling function
  • Distribution functions of discrete probability distributions
Simple function Disjoint sets
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