Monday, October 25, 2010


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.


  1. Possibly true, but if we break things down to the (sub)atomic level, there may be discrete values rather than continuity. Perhaps everything is a piecewise function, but the pieces are small enough as to be construed as continuous on a macro- scale.

    We just got done talking about sketching derivatives given a graph of a function in calculus. It's hard to think of "actual" physical reasons for things like jump discontinuities or infinite asymptotes without going to some odd topology, so I tell the students to think of things like "warp gates" in video games and how you might run into one at one velocity but come out at another in another direction.

    Anyways, yeah, it is funny how we (mathematicians) spend forever trying to iron out the little "details" that are so hard to come to terms with (Cantor's set, for another example) when they seem to have very little real life application.

  2. Yeah, I that's why I said to my knowledge, because currently we haven't gotten down that far under the microscope, but I agree with you about discrete values. It probably will turn out that it is that way or using completely new sets of physical laws or that a new, more accurate math is discovered.

  3. Some physicists seem to think that the planck time is actually the theoretical minimum time interval and that time is then passing in discrete "frames". I don't know how much legitimacy other scientists perceive in that theory, but there's definitely not a consensus.