R J Lipton has an engaging post which starts from something that rather horrified the mathematics community when it was discovered: it’s a function which is continuous everywhere, but it’s not differentiable anywhere, no matter where you look. Continuity and differentiability are important concepts in mathematics, and have very precise definitions — motivated, in part, by things like the difficult function Lipton discusses here — but they can be put into ordinary language tolerably well.
If you think of a continuous function as being one whose graph you could draw without having to lift the pencil from the paper you’re not doing badly. Similarly a function is differentiable at a point if, from that point, you know what way the curve is going. This function, found by Karl Weierstrass, is one example of the breed.
Lipton points out the somewhat unsettling point that it’s much more common for functions to be like this than they are to be neat and well-behaved functions like
or even
, in much the same way a real number is much more likely to be irrational than it is to be rational, and Lipton goes on to give an example in an area of mathematics I’m not familiar with of the “pathological” case being the vastly more common one. Fortunately, it turns out, we can usually approximate the “pathological” or “monster” function with something easier to work with — a very fortunate thing or we could get done very few computations that reflected anything actually interesting — and that’s another thing we can credit Weirstrass with discovering.
Karl Weierstrass is often credited with the creation of modern analysis. In his quest for rigor and precision, he also created a shock when he presented his “monster” to the Berlin Academy in 1872.
Today I want to talk about the existence of strange and wonderful math objects—other monsters.
Weierstrass’s monster is defined as
$latex displaystyle f(x) = sum_{k=1}^{infty} a^{k} cos(b^{k}pi x), &fg=000000$
where $latex {0<a<1}&fg=000000$, $latex {b}&fg=000000$ is any odd integer, and $latex {ab>1+3pi/2}&fg=000000$. This function is continuous everywhere, but is differentiable nowhere.
The shock was that most mathematicians at the time thought that a continous function would have to be differentiable at a significant number of points. Some even had tried to prove this. While Weierstrass was the first to publish this, it was apparently known to others as early as 1830 that such functions existed.
This is a picture of the function—note its recursive structure, which is…
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