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Step Functions

A step function is a piecewise-defined function in which every piece is a horizontal line segment or a point.

Example 1:

Let the function shown be defined for all the integers as

y = 2 for x < 1 y = 3 for x 1

This function is made up of infinitely many discrete points each of which have a y -coordinate of either 2 or 3 .

Example 2:

Consider the function, y = n for n < x n + 1 where x , n .

This function is made up of infinitely many horizontal segments, each of which includes the right endpoint but not the left endpoint.

Example 3:

y = 1 for x < 0 y = 0 for x = 0 y = 1 for x > 1 .

This step function is called the Heaviside Function. It may look very simple, but it is extremely important in electrical engineering and control theory.