A Pathological Function


WE DEFINE a function F(x) which takes every real value in every interval.

Express the fractional part of x as a binary "decimal" (which may or may not terminate) 0.x1x2x3 ...  ..  ..  ..  (1)

We denote the sequence 0111 ... (n 1's)0 by Sn and define

fn = 0 if Sn occurs a finite number of times in (1)
fn = 1 if Sn occurs an infinite number of times
andf(x) = 0.f1f2f3 ... (another binary decimal).

Now take any interval of x (as small as we like). We can find N.x1x2...xr such that with these fixed x is bound to lie in the interval. We can now make f(x) take any value 0.f1f2... between 0 and 1. For consider the sequence n1,n2,n3, ... (which may terminate) of all the n's for which fn = 1, and write

x = N.x1x2...xr; Sn1; Sn1, sn2; Sn1, Sn2, Sn3; ...

which contains all Sni an infinite number of times and all other Sn a finite number of times. By considering F(x) = tan \pi {f(x) - ½} we have a function which takes every real value in every interval of x.

Eureka, 20.

Reproduced from Eureka 27 page 31.
HTML conversion Copyright © 2002-4 The Archimedeans.

Additional notes to the online version

Errata: Sn1, Sn2, Sn3, Sni should read Sn1, Sn2, Sn3, Sni (multiple places).

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