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Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all $x,y,m\in\mathbb{Z}$ $$x\equiv y \pmod m \implies f(x)\equiv f(y) \pmod m.$$ Or in other terms $$f(x) \text{ mod } m = f(x \text{ mod } m) \text{ mod } m$$ My question is

  1. Does this property have a name?
  2. One example of such functions are polynomials with coefficients in $\mathbb{Z}$. Are there any other examples? Is there some classification?
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    You can rephrase the condition as $x-y \lvert f(x)-f(y)$ for all $x,y\in\mathbb{Z}$ – 2017-01-13
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    Is $m$ fixed or not? – 2017-01-13
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    its not. It should hold for all $m$. – 2017-01-13
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    This is the same thing as saying that the function $f:\mathbb Z_m \to \mathbb Z_m$ is well-defined. – 2017-01-13
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    https://www.irif.fr/~seg/_2015SLIDES_Yurifest_Berlin/2015YURIFEST_Berlin.pdf contains many examples and a characterization. They call them "congruence preserving" functions – 2018-05-08
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    @m_t_ Yes, thats exactly what I was looking for. Thanks. – 2018-05-09
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    You're welcome. Thank you for posting such an interesting problem. – 2018-05-09

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