Is it possible to construct a function such that it alternates between two values after each "infinitesimal"? Say $0$ and $1$. Now I am not able to properly define the question, but in terms of infinitesimals, what I want is that if $f(x)=0$, then $f(x+dx)=1$ and $f(x+2dx)=0$ and so on. (Maybe someone can help define the problem rigorously too?)
Like there is a function which is $0$ for rationals and $1$ for irrationals. But it's not exactly alternating in the sense I want. Plus, irrationals are way more dense than rationals, so this obviously doesn't work.
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Aditya Agarwal
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1There are no “infinitesimals” in $\mathbb{R}$. – Vivaan Daga Feb 27 '22 at 07:50
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@VoiletFlame, yes I know that. That's why I said I couldn't define it rigorously. – Aditya Agarwal Feb 27 '22 at 07:55
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Reminds me of the ito derivative – tryst with freedom Feb 27 '22 at 08:17
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2I don’t understand, what you want and why, you could let $dx$ be a really small real number. – Vivaan Daga Feb 27 '22 at 08:36
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The "alternating" business is impossible to satisfy since there's no such thing as a "next real number", so I'm going to ignore that part.
But maybe we can fix the irrationals and rationals example by using a set that fills exactly half of each subinterval. Unfortunately, for measure theory reasons typically covered in an upper undergraduate or graduate level real analysis course, even that is impossible. (This question has a proof and is marked as a duplicate of this more general question.)
What about weakening the request even further. Is there a set that takes up half of the interval and is at least kind-of spread out? That's vague, but one thing that comes to mind is the Smith-Volterra-Cantor set which is nowhere dense (its set of limit points doesnt contain any interval). Unfortunately, its complement contains a bunch of intervals.
Is there a pair of sets that each take up half of the whole interval $[0,1]$, neither of which contains any intervals and are kind of spread out (so "rationals on the left half, irrationals on the right half" doesn't count)? I don't know, but maybe a version of that is worth asking as a separate question if it interests you. (Note, we can't require them both to be nowhere dense.)
Mark S.
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