Final answer:
There seems to be a confusion about the greatest integer function. For the greatest integer function to be greater than 11 or equal to 12, x must be in specific intervals. In probability, if a function is constant and normalized over an interval, the probability over the entire interval is 1.
Step-by-step explanation:
The greatest integer function, often denoted by modulus of x or ⌊ x ⌋, assigns to any real number x the largest integer less than or equal to x. When the question mentions that the function f such that f(x) is equal to modulus of x is greater than 11 or 12, it seems there might be a misunderstanding in the interpretation of the greatest integer function.
If we are asked to prove that the greatest integer function is greater than 11 or 12 for a certain domain, we would look at the range of x values that make the floor function ⌊ x ⌋ greater than 11 or equal to 12. Specifically, for ⌊ x ⌋ to be greater than 11, x would need to be greater than 11 but not an integer, since the greatest integer less than or equal to x would be 11 itself.
For ⌊ x ⌋ to be equal to 12, x would need to be in the interval [12, 13), since the greatest integer less than or equal to any number between 12 and 13 non-inclusive is 12.
However, if the context is probability and we are looking for the probability P(0 < x < 12) of a continuous probability function that equals 12 for 0 ≤ x ≤ 12, it implies f(x) is a constant function. Assuming the function is normalized over this interval, P(0 < x < 12) would be equal to 1 since the probability covers the entire interval for which the function is defined.