Test floor function on negative numbers:
(*assumefractionpartofnegativexisϵ,whichisin(-1,0]*)​​FractionalPart[-5.7]
Out[]=
-0.7
(*assumek=floor[x]fornegativenumber*)
In[]:=
Floor[-5.7]
Out[]=
-6
(*thenumeratoris-2k-1-ϵ*)
In[]:=
5.7-Floor[-5.7]
Out[]=
11.7
Ranges for each interval for k: (-1,0], (-2,-1], (-3,-2] ...
In[]:=
gl=Table
-2*k-1
2-k
,
-2*k
2-k
,{k,-7,0}
Out[]=

13
3
,
14
3
,
11
2
2
,3
2
,
9
7
,
10
7
,
7
6
,4
2
3
,
5
,
6
5
,
3
2
,2,
1
3
,
2
3
,-
1
2
,0
Test if the intervals are disjoint: lower for next is larger than the upper bound of the previous interval:
In[]:=
Reduce
-2*(k-1)-1
2-(k-1)
>
-2*k
2-k
&&k<=-1,k,Integers
Out[]=
k∈&&k≤-1
In[]:=
Plot
Abs[x]-Floor[x]
2-Floor[x]
,{x,-6.5,0},GridLines->{Range[-6,0],Join@@gl},GridLinesStyle->Directive[{Thickness[0.002],Dashed,Opacity[0.5]}]
Out[]=