Re: monotone increasing function on the reals taking only irrational values

In article <gerry-1770DC.12433215112006@xxxxxxxxxxxxxxxxxx>,
Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <1163551446.220794.261370@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
dmadden@xxxxxxx wrote:

I ran across a question as to whether such a function exists and I
can't think of any obvious way to prove it can't. Indeed I think you
could construct such a function if you had a convergent series of
positive terms such that the sum of any infinite subsequence is
irrational. Is there such a thing? Like maybe if the terms converge
very very quickly towards zero????

How about this:

Let the binary expansion of x be x = 0 . a_1 a_2 a_3 a_4 ....

Then let f(x) = 0 . a_1 0 1 a_2 0 0 1 a_3 0 0 0 1 a_4 0 0 0 0 1 ....

I think this does the job on [0, 1)
and is easy to extend in various ways to all of R.

Nice, that's a lot easier than what I did.
.

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