Introduction:
The function f(x) = (-1)^(⌊x⌋), where ⌊x⌋ denotes the greatest integer less than or equal to x, is the function in question. We need to determine whether this function is odd or even.

Odd Function:
An odd function is defined as a function that satisfies the property f(-x) = -f(x) for all values of x in the domain of the function. Let's examine whether f(x) satisfies this property.

Proof:
To prove that f(x) is an odd function, we need to show that f(-x) = -f(x) for all x in the domain of f(x).

Case 1: x is an integer:
If x is an integer, then ⌊x⌋ = x. Therefore, f(x) = (-1)^x.

For f(-x), we have ⌊-x⌋ = -x since -x is the greatest integer less than or equal to -x. Therefore, f(-x) = (-1)^(-x).

Now, let's compare f(-x) and -f(x):

f(-x) = (-1)^(-x)
= 1/((-1)^x) (since (-1)^(-x) = 1/((-1)^x))
= 1/((-1)^x)
= -1^x (since 1/(-1^x) = -1^x)

Since f(-x) = -f(x), we have shown that the function is odd when x is an integer.

Case 2: x is not an integer:
If x is not an integer, then ⌊x⌋ is the greatest integer less than x. Therefore, f(x) = (-1)^(⌊x⌋).

For f(-x), we have ⌊-x⌋ = -x + 1 since -x + 1 is the greatest integer less than or equal to -x. Therefore, f(-x) = (-1)^(⌊-x⌋) = (-1)^(-x + 1).

Now, let's compare f(-x) and -f(x):

f(-x) = (-1)^(-x + 1)
= -1 * (-1)^(-x) (since (-1)^(a+b) = (-1)^a * (-1)^b)
= -f(x)

Since f(-x) = -f(x), we have shown that the function is odd when x is not an integer.

Conclusion:
In both cases, we have shown that f(-x) = -f(x) for all values of x in the domain of the function, which satisfies the definition of an odd function. Therefore, we can conclude that f(x) = (-1)^(⌊x⌋) is an odd function.