The correct options are
A f is an even function when x∈Z, f is an odd function when x∉Z
C f is an even function for all integers
D Neither odd nor even
Let f:R→R be a function defined as f(x)=(−1)[x].
When x is an integer [x]=x, hence f(x)=(−1)[x]=(−1)x, which is an even function.
If x is a positive real number which is not an integer, then [x] is the integer part of the given number, say a.
Therefore f(x)=±1 based on a is even or odd.
If x is a negative real number which is not an integer, then [x]=a−1, where a is the integer part of the given number.
Therefore f(x)=±1 based on a is odd or even.
Hence f(x)=−f(−x) when x∉Z