What are some simple examples of functions $f,g\colon \mathbb Z\to\mathbb Z$ which are bijective and their sum is also bijective?
Unless I am missing something, it is not difficult show that such functions exist by induction. (We simply order $\mathbb Z=\{z_n; n=0,1,2,\dots\}$ in some way. And we can define $f$, $g$, $h$ by induction in such way that we make sure that after $n$-th step: a) $f$ and $g$ are defined for the integers from some set $A_n$; b) $f$, $g$ and $h=f+g$ are injective on $A_n$; c) $z_n\in A_n$; d) $z_n$ belongs to $f[A_n]$, $g[A_n]$, $h[A_n]$. If needed, I can try to make this more precise and post inductive construction in an answer; of course, if somebody wants to post such an answer, you're more than welcome to do so. Especially if you can suggest some more elegant way than what I sketched here.)
But I have doubts that there is a bijection given by a simple formula. (Although I admit that the words "simple formula" are rather vague.)
So, as an additional question, is there example of bijections $f$, $g$ such that $f+g$ is also bijection, and these functions are "nice"? (For some reasonable meaning of the word "nice".) Or can we show that such example cannot be found if we restrict $f$, $g$ to some reasonably behaved class of functions?
Basically the motivation for this question came from a course that I am TA-ing. As an exercise, to help them acquainted with the notion of bijection, the students were asked to find an example of two bijections from $\mathbb Z$ to $\mathbb Z$ such that their sum is not a bijection. A colleague, who is also TA at the same course, asked me whether I can think of example where the sum is bijection, since for the most natural examples that one typically thinks for (such as $x\mapsto x+d$ or $x\mapsto -x+d'$) you never get a bijection.
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