Given a set $A \subseteq R$,$i \in Z$ ,$A_i=A+i$, $\bigcup_i A_i=R$,{$A_i$} pairwise disjoint, is $A$ necessary an interval?
$A+i$ means set $A$ has all it's elements translated by an amount $i$.
The result is obviously true if $A$ is an interval for example $A=(0,1]$. But is it always the case?
Consider $A=(0,1)\cup \{2\}$. Then you can see $ A_i\cap A_j = \emptyset $ for $i\neq j$ and $\bigcup_i A_i=\mathbb R$.
But you can do it more extreme. For example, you can divide $[0,1)$ in countable subintervals and shift each by different integers.
In addition, you can choose a nowhere connected set. But that is a little bit more technical.