Step function

A step function or staircase function is a piecewise constant function having only finitely many pieces. It is a function that increases or decreases abruptly at each step without any smooth transitions between values. Step functions are commonly used in mathematics, especially in the study of integral calculus, signal processing, and probability theory.

Definition

A function f: ℝ → ℝ is called a step function if there exists a finite subdivision of the real line, {x_0, x_1, ..., x_n}, such that f is constant on each interval (x_i, x_{i+1}) for i = 0, 1, ..., n-1. The values at the endpoints of the intervals can be defined arbitrarily to include or exclude the endpoints.

Examples

One of the most well-known examples of a step function is the Heaviside step function, H(x), which is defined as 0 for x < 0 and 1 for x ≥ 0. Another example is the rectangular function, which is similar to the Heaviside function but defined over a finite interval.

Applications

Step functions are widely used in various fields of science and engineering. In integral calculus, they serve as simple examples of functions that are easy to integrate. In signal processing, step functions are used to model signals that switch on and off abruptly. In probability theory, step functions are used to describe distribution functions of discrete random variables.

Properties

Step functions have several important properties. They are bounded, since there is a finite number of steps, each of which has a finite height. They are also measurable, which means they can be integrated, and their integral is equal to the sum of the areas of the rectangles defined by the steps.

See also

• Function (mathematics)
• Piecewise
• Heaviside step function
• Rectangular function