Step functions are functions which change in value from one point to another. The function may take a certain value on an interval, and at each end of the interval the value of the function may change quite suddenly to some other value.
Though the step function changes suddenly at certain points, it must be well defined in the sense that the function cannot have two values at a particular point, so that the function is defined at
by the function on the interval
or for the function on the interval
but not both.
Step functions arise quite naturally in many practical situations. Many car parks charge per part of an hour:
0 – 1 hours £1
1 – 2 hours £2
2 – 3 hours £3
and so on. At the end of each hour you have to put another pound in the meter to remain in the car park and you cannot put in a few extra pence to stay another few minutes.