Typewriter sequence on R

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I'm familiar with the typewriter sequence on the unit interval, but I'm wondering if there is a way to extend it to the entire real line. If there is, could someone explain how to make such an extension or provide a formula for such a sequence?

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You can divide $[-N,N]$ into intervals of length $1/N$ (or $2^{-N}$), and increase $N$ to infinity.