Monday, October 25, 2010

Cusps

It occurred to me a couple of days ago when I was doing some Numerical Analysis homework how such a large part of mathematics, calculus in particular, depends on the assumption that a function is a smooth, continuous, differentiable function, and how a great deal of math breaks down at a cusp. A cusp, a theoretical point where a function makes a "sharp" turn, to my knowledge, only exists in pure mathematics. Nowhere in real life does a point (in the classical, pure mathematical definition) exist to our knowledge (i.e. we have not observed a point of zero area). Exceptions, such as the center of a black hole, break down known laws about the universe much in the same way that cusps break down many laws of mathematics. This thought came about when I was trying to fit polynomials of various degrees to a piecewise function that contained within the domain an absolute value term. I know it may not seem that incredible to many of you, but to me this is profound. Something as simple as a slight deviation from a smooth curve is impossible to differentiate. I don't have much to say about the subject, but this post was simply to give the reader something to think about.