Do Functions Have to Be Continuous Discrete Mathematics
One thing that I frequently touch on casually as I'm writing this blog is the distinction between continuous mathematics, and discrete mathematics. As people who've been watching some of my mistakes in the topology posts can attest, I'm much more comfortable with discrete math than continuous.
The distinction is a very important one. Continuous mathematics is, roughly speaking, math based on the continuous number line, or the real numbers. The defining quality of it is that given any two numbers, you can always find another number between them - in fact, you can always find an infinite set of numbers between them. Building up from numbers, a function in continuous math can take the form of a perfectly smooth curve without any gaps or breaks. In theory, you can talk about a function as a set of pairs - f={(x,y) | y=f(x)} - but you can't even show an exhaustive list of the pairs that make up the function, even over a finite section of the function.>
In discrete mathematics, you're working with distinct values - given any two points in discrete math, there aren't an infinite number of points between them. If you have a finite set of objects, you can describe the function as a list of ordered pairs, and present a complete list of those pairs.
The difference becomes clearer when you think about some of the deeper areas of math - which is sort of unusual. In general, getting deeper makes things harder; but here, getting deeper makes the difference easier to understand.
In continuous math, the fundamental set of numeric values that we use for proofs is the interval (0,1). We often prove various properties of sets by using mappings from values in the range (0,1). In discrete math, the fundamental set of
numeric values is the natural numbers, we prove properties of sets by using mappings from the natural numbers.
For example, one of the basic fundamental sets of concepts in math consists of
the ideas of shape, closeness, and adjacency:
- In continuous math, we generally
study those ideas using topology: sets of points form topological spaces, and we often study properties of those spaces and functions over them
by using mappings from the range (0,1) to subspaces or functions. - In discrete math, we study those ideas using graph theory, where
we have a set of points where each point is connected to a specific set of
other points by edges. We often study properties of graphs or functions over graphs using mappings from the natural numbers to subgraphs or functions.
Another very fundamental thing we do in math is study how things change. In continuous math, we do that using differential equations, which are functions that describe the rate of change of one function using another derived function. In discrete math, we do the same thing using recurrence relations, which define a the value of a function at a point in terms of one or more of the points that precede it.
For example, in continuous math, given the equation y=x2, we can say that at any x, y is changing at a rate of 2x. In discrete math, we can describe a set like the fibonacci series by saying fib(x)=fib(x-1)+fib(x-2).
I'll close this with a personal story about differential equations and recurrence relations. Back in college, I started school as an electrical engineering major. I was a very bad electrical engineer, and I wound up basically flunking out. One of the courses that I failed in my last semester as an EE was differential equations. After that semester, I took a year off to get my head together and figure out what I wanted to do. When I came back, I decided to take differential equations again - not because I needed to, but because I wanted to prove to myself that I could do it.
For roughly the first half of the semester, I struggled. I worked my tail off, and barely managed to scrape by with Ds on my exams.
At the same time, I was taking the discrete math course required for computer science students. Around the midpoint of the semester, we we studying recurrence relations, and how to translate them into closed form - that is, a non-recurrence based equation.
One of my closest friends and I were taking both classes together, and one afternoon, we were working together on our homework in both classes. We started with diffEQs, and I got incredibly frustrated, and so we put it down, and switched to
the recurrence homework. And were zipping through the recurrence problems like nobodies business, just knocking 'em down. And as we were doing this, we noticed (or to be honest, I think she noticed) that one of the problems was almost exactly the same as one of the problems in our diffEQs homework, and that the way that I had just solved it was almost exactly the way that you'd solve the diffEQ. And I looked at it, and looked at it... And she was absolutely right. They were really pretty much the same thing: that recurrence relation was the discrete form of a differential equation. And it clicked. And from that afternoon on, I never had any more trouble with diffEQs. I ended up aceing the rest of the exams for the class, and finished with a B.
The point of that is partly how great the similarity between differential equations and recurrence relations really is; and partly to show you just what you can do to yourself when you convince yourself that you can't do something. There was nothing about differential equations that I couldn't do. I won't say it was easy, but it was always, from day one, entirely within my ability to understand it and do it. But I was convinced that it was so hard that I'd never grasp it. And so I didn't. I really believed that I was trying to - but on some level, I was so sure that I couldn't do it that I wouldn't let myself understand. Until my friend hit me in the face with the fact that I could do it, and in fact, already had done it.
(And as a personal aside - that friend is still one of my best friends. She was the "Best Man" at my wedding; I'm the godfather of her first son. If you're reading this Abby, I don't think I ever really thanked you for that blow to the head in the lecture hall at Lorree. So thanks!)
Source: https://scienceblogs.com/goodmath/2007/03/01/basics-discrete-vs-continuous-1
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