# 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.*x*_{1}*x*_{2}*x*_{3} ... .. .. .. (1)

We denote the sequence 0111 ... (*n* 1's)0 by
*S*_{n} and define

| *f*_{n} = 0 if *S*_{n} occurs a finite number of times in (1) |

| *f*_{n} = 1 if *S*_{n} occurs an infinite number of times |

and | *f*(*x*) = 0.*f*_{1}*f*_{2}*f*_{3} ... (another binary decimal). |

Now take any interval of *x* (as small as we like). We
can find *N*.*x*_{1}*x*_{2}...*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.*f*_{1}*f*_{2}... between 0
and 1. For consider the sequence
*n*_{1},*n*_{2},*n*_{3}, ... (which may
terminate) of all the *n*'s for which *f*_{n} =
1, and write

*x* = *N*.*x*_{1}*x*_{2}...*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**.

Reproduced from Eureka 27 page 31.

HTML conversion Copyright © 2002-4 The Archimedeans.

## Additional notes to the online version

Errata: *S*_{n1}, *S*_{n2},
*S*_{n3}, *S*_{ni} should read
*S*_{n1}, *S*_{n2},
*S*_{n3}, *S*_{ni} (multiple places).

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Online HTML version last updated:
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