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Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure

$$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$

Then

$$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + 1 \mp \cfrac{1}{a_2 + 1 \mp \cdots}}.$$

That is, replace each instance of "$a_i +$" with "$(a_i + 1) -$" and of "$a_i -$" with "$(a_i + 1) +$". See here for the motivation behind the following questions.

Questions

I know that $f$ is a function, but I can't type this definition into a graphing calculator directly (that I know of).

  1. How can this function be written in a "well-formed" way, without english and with the ability to type it directly into a graphing calculator?
  2. Do there exist functions that can't be written in a "well-formed" way? What are they called?
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  • $\begingroup$ Do you want to have the $b_i$ there? Continued fractions become nonunique when they are introduced, which risks the function property. I do not know if the represented form matters, but it seems like it might. $\endgroup$ Jan 2, 2016 at 19:46
  • $\begingroup$ Do we have any guarantees on the $b_i$'s? Is it the case that all $b_i$ in a nearest integer continued fraction for an $0 \lt x \lt 1$ are equal to $1$? If so, I'll change the question to reflect. $\endgroup$ Jan 2, 2016 at 20:07
  • $\begingroup$ It is the case that every irrational has a unique representation such that $b_i=1$. If $b_i\neq 1$ then there are multiple continued fraction representations of the same number $\endgroup$ Jan 2, 2016 at 20:09
  • $\begingroup$ I modified the question by replacing the $b_i$'s with $1$'s in order to ensure the function property. $\endgroup$ Jan 2, 2016 at 20:13

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