A Pathological Function

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A Pathological Function

BY J. D. ROBERTS

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 S_n and define

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

Now take any interval of x (as small as we like). We can find N.x₁x₂...x_r 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 f_n = 1, and write

x = N.x₁x₂...x_r; S_n1; S_n1, s_n2; S_n1, S_n2, S_n3; ...

which contains all S_ni an infinite number of times and all other S_n a finite number of times. By considering we have a function which takes every real value in every interval of x.

Eureka, 20.

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Reproduced from Eureka 27 page 31.