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.
17 thoughts on “A Summer 2015 Mathematics A To Z: jump (discontinuity)”
I think the main reason we like continuous functions is because they are defined at every value. In physics, discontinuities and singularities in functions that are equations of motion give us domains where the theory is no longer valid.
That continuous functions are defined at every value, or at least over the wide range of values, is certainly part of why we like them. And you can, to an extent, describe most real-world phenomena with continuous functions. Even, say, the flow of electricity through a switch that’s been turned on can be represented fairly as a steeply-rising but never actually discontinuous function.
But there’s a bit of selection bias here. We have very good tools for studying continuous functions, and better ones for differentiable and smooth functions, so we do try to model problems as such. If we had better ways of understanding discontinuous functions we’d surely represent more phenomena as sudden jolts and shocks. After all, we do exactly that when we have the tools to analyze that. If we want to model a mass on a damped spring receiving a shock, we always model the shock as an instantaneous thing, rather than the fast but continuous push it really receives.
any physical quantity can only be represented by a continuous function. A shock or jolt to a mass on a spring is still continuous, it just has a beginning. The force is still well defined throughout time, it’s just an infinitesimal force over infinitesimal time for tiny intervals. If you are thinking of more advanced analysis using dirac delta functions and wave packets, those do not represent physically realizable states. We have to stitch together discontinuous functions like dirac deltas into wave packets to get wavefunctions that represent real physical quantitites. Electricity flowing through a switch is completely continuous, electrons are moving a finite amount of space during a finite amount of time always. Even the wave functions that represent the electrons in Quantum Electrodynamics are fully continuous, though they do have singularities for weird values. If a function is discontinuous, by definition that means it is not defined for a certain value. If I ask what is the force at time t and you say I don’t know, we are throwing that theory out.
Yeah, locality as it’s called is a key feature of all theories of nature we have to this day. Basically, energy, momentum, any conserved quantity can only move through a finite amount of space in a finite amount of time, or in other words its derivative is always well defined and its movement is always continuous.
I am looking forward to your account of Catastrophe Theory ! Coming soon ?
Thank you most kindly. I’m afraid I know so little about catastrophe theory I’d be in poor shape explaining any of it, though.
I have considered doing a sequence in which I plunge into a field I’ve wanted to learn better and report what I learn as I learn it. That might risk being impenetrable to the normal reader, though.
I cackled when I first saw the function that’s continuous everywhere and differentiable nowhere. It was almost as impressive as turning one unit sphere into two unit spheres with just translation and rotation.
Those are fun things, certainly. They’re kind of hard things to grasp, though; you need a good running start to get to either topic and there are a lot of points where intuition protests. I think more accessible is the function that’s discontinuous everywhere except at a single point. But even that looks almost like mathematicians showing off how they can be tricky.
Some discontinuous functions look like fractions with a polynomial in the numerator and a different one in the denominator. People learned to hate fractions at an early age so they will often hate these rational functions, too.
Oh, yes, yes. I had forgotten rational functions, somehow. I say somehow because I do remember spending roughly twenty-six years during my senior year in high school learning various tricks for integrating rational functions. I suppose I’ve used them (rational functions) since then, some. Integration tricks for them … well, I guess in a few differential equations courses, but not much for all the time spent learning how to do that.