Sunday, March 6, 2011
Sunday Numbers 2.0, Vol. 3:
It's easy to explain the idea of a continuous curve on a surface in everyday English. If you can draw a picture without ever lifting the pen from the surface, that is a continuous curve.
In this picture, let's assume the blue curve is just being hidden by the red curve at the five points of intersection. This means we can see the red picture is continuous and we will assume the blue picture is continuous. The difference is that the blue line represents a "smooth curve" and the red does not. This becomes important in differential calculus.
This particular drawing is from a lesson on how integral calculus works to find the area under the blue curve and bordered below by the horizontal line (representing the x-axis) and between the vertical lines labeled a and b. The graph in red is also a continuous curve.
There are lots of ways to have mathematical functions that create discontinuous graphs as well. The function pictured to the left is y = Floor(x), represented in most math books by the odd looking brackets you see in the picture where the bottom of the brackets exist but the top do not. The idea of Floor is that any real number x has a closest integer that is less than x. For example, Floor(2.5) = 2 and Floor(pi) = 3. (Oh-h-h-h-h-h... Floor(pi)!) The Floor of any whole number is itself, so Floor(4) = 4, but as soon as we move down from 4 the tiniest tick, Floor(3.999999999) = 3. This means you have to lift the pen off the surface and put it down away from the line segment you were drawing previously.
The standard way for mathematicians to state that a function is continuous this is "The function f(x) is continuous at all points x for which it is defined". the converse is "The function g(x) is not continuous at some set of points". It can also happen that a function may not be defined at a point x, but can either be continuous for all points near x, which we call a neighborhood of x. Conversely, a function may be undefined at a point and discontinuous at that point. A famous example of that kind of discontinuity is y = 1/x, which isn't defined at 0, is approaching infinity if x is positive and close to zero, but approaching negative infinity is x is negative and close to zero.
The mathematical template for continuity proofs is called epsilon-delta, where these two Greek letters are stand-ins for really small numbers. In the picture to the left, the red line represents the function f and f(a) = b. If we want to prove that f is continuous at a, which the picture shows to be true, the plan of attack is to let some nebulous observer choose a small number we will call episilon. What is asked of us is to find a neighborhood around a such that the inequality b - epsilon < f(x) < b + epsilon for every x in the neighborhood. We usually are asked to make the neighborhood around a to be symmetrical around a so the letter delta is added to a and subtracted from a to give us the boundaries of the neighborhood.
Let me give an example. We want to prove f(x) = x² is continuous around x = 4. I could let epsilon equal some specific small number like 0.01, but the true proof comes from proving it for any given small number. Since 4² = 16, we would look for the square roots of 16+epsilon and 16-epsilon, both of which will be very close to 4 is epsilon when very small. We then would see which square root is farthest from 4, and the distance would be our delta.
This is a very standard part of analysis. The standard "big" way to split up math is into analysis and algebra, and if I had my druthers, I'd rather do algebra. A lot of proofs in analysis are kind of tedious, while occasionally, algebraic proofs can be very pretty and elegant. If you want to see the average math graduate student's eyes glaze over and a nearly unstoppable desire to sleep overtake him or her, just say "epsilon-delta".
I don't know, someone from the analytical side of the field might stumble upon this post and give a spirited defense of epsilon-delta proofs, but I have to admit, if it happens, it will be the first time I've heard of it.
Next week: Logic and eccentric Britons.