The Sunday Function

I sometimes forget that we don’t all read the same blogs, and that it’s good to recommend some of the fun stuff out there on the internets. So let me give a shout-out to Matt Springer at Built on Facts, who had the brilliant idea of discussing a different function every Sunday. Functions are one of those things that are as necessary to math and science as breathing, but which don’t necessarily percolate into the wider world. And he (quite correctly, I think) interprets his self-imposed mandate fairly liberally, taking the time to talk about various issues in middle-level mathematics. Here are some selections from Matt’s series:

Consider this an open thread to recommend other stuff we should all be reading. Or your favorite functions.

11 Comments

11 thoughts on “The Sunday Function”

  1. Here is mine:

    g(x) = exp(-1/x^2) for x not equal to zero
    = 0 if x = 0

    g is “C-infinity” in that it has derivatives of all orders everywhere but is not analytic at x = 0, which means it doesn’t have a Taylor series about x = 0 that is valid on any open interval containing zero.

    This is the basic building block of the “bump function” that allows us to do “surgery” in differential topology but the stumbling block between extending results from smooth to analytic manifolds.

  2. Pingback: A Cold Walk, A Lamarckian Result, Conservative Lust for Porn and other topics « blueollie

  3. And then there’s Ackermann’s function, which I covered on february 10th, and the devil’s staircase which I’ll cover sometime this month. The latter has a zero derivative almost everywhere, except at the cantor dust points where it is infinite. It climbs from 0 to 1 ONLY at the cantor points. Continuous but not differentiable 😉

  4. My favourite is (does it have a name?):

    f(x) = 1 /(1 – e(-1/x))

    lim f(x) = 1
    x -> 0+

    lim f(x) = 0
    x -> 0-

  5. Gavin: that is what as known as the “bump function” ; it is used in differential topology to “sew” things together.

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