The function f(x)=sec−1⁡xx−[x], where [x] denotes the greatest integer less than or equal to x, is defined for all x∈

The function sec−1⁡x is defined for all x∈R−(−1,1) and the function 1x−[x] is defined for all x∈R−Z. So, the given function is defined for all x∈R−{(−1,1)∪{n∣n∈Z}}.