A Summer 2015 Mathematics A To Z: jump (discontinuity)
Analysis is one of the major subjects in mathematics. That’s the study of functions. These usually have numbers as the domain and the range. The domain and range might be the real numbers, or complex numbers, or they might be sets of real or complex numbers. But they’re all numbers. If you asked for an example of one of these functions you’d get something that looked more or less like a function out of high school.
Continuity is one of the things mathematicians look for in functions. To a mathematician continuity means almost what you’d imagine from the everyday definition of the term. You could draw a sketch of a continuous function without having to lift your pen off the paper. (Typically. If you want to, you can define functions that meet the proper mathematical definition of “continuous” but that you really can’t draw. Mathematicians use these functions to keep one another humble.)
Continuous functions tend to be nice ones to work with. Continuity usually makes it easier to prove a function has whatever other properties you’d like. Mathematicians will even talk about continuous functions as being nice and well-behaved and even normal, as though the functions being easier to work with bestowed on them some moral virtue. However, not every function is continuous. Properly speaking, most functions aren’t continuous. This is the same way that most numbers aren’t whole numbers.
There are different ways that a function can be discontinuous. One of the easiest to understand and to work with is called a jump discontinuity. If you draw a plot representing a function with a jump discontinuity, it looks rather like the plot of a nice, well-behaved, continuous function except that at the discontinuity it jumps. From one side of the discontinuity to the other the function suddenly hops upward, or drops downward.
If a function only has jump discontinuities we aren’t badly off. We can write a function with jump discontinuities as the sum of a continuous function and a function made up only of jumps. The continuous function will be easy to work with, since it’s continuous. The function made of jumps isn’t continuous, by definition, but it’s going to be “flat” — it’ll have the same value in-between any two jumps. That’s usually easy to work with, and while the details of these jump functions will be different they’ll all look about the same. They’ll have different heights and jump up or down at different points, but if you know how to understand a function that jumps from being equal to 0 to being equal to 1 when the input goes from just below to just above 2, then you know how to understand a function that jumps from being equal to 0 to being equal to 3 when the input goes from just below 2.5 to just above 2.5.
This won’t let us work with every function. Most functions are going to be discontinuous in ways that we can’t resolve with jump functions. But a lot of the functions we’re naturally interested in, because they model interesting problems, can be. And so we can divide tricky functions into sets of functions that are easier to deal with.