The correct options are
A limx→nf(x) exists
B limx→kf(x) exists
D f is continuous at x=k
f(n)=[n]+[−n]=n−n=0
limx→n−f(x)=limh→0+[n−h]=[−n+h]=n−1+(−n)=−1
limx→n+f(x)=limh→0+[n+h]=[−n−h]=n+(−n−1)=−1
If k∉I, the function [x] and [−x] are continuous at k.
Hence limx→k f(x) exists.