f(x) = [[x]] is the "greatest integer" function. The range of f(x) = x - [[x]] is:

To determine the range of $f(x) = x - [[x]]$, we need to understand the behavior of the greatest integer function, denoted as $[[x]]$, which returns the largest integer less than or equal to $x$.

1. Definition and Properties:

   • For any real number $x$, $[[x]]$ is an integer.
   • $x - [[x]]$ represents the fractional part of $x$.

2. Range Analysis:

   • Let $x$ be any real number.
   • $[[x]]$ is the integer part of $x$.
   • Therefore, $x$ can be written as $x = n + f$, where $n$ is an integer ($[[x]]$) and $f$ is the fractional part such that $0 \leq f < 1$.

3. Expression Simplification:

   • $f(x) = x - [[x]]$
   • Substituting $x = n + f$, we get $f(x) = (n + f) - n = f$.

4. Range of $f$:

   • Since $f$ is the fractional part of $x$, it ranges from 0 (inclusive) to 1 (exclusive).
   • Therefore, $0 \leq f(x) < 1$.

5. Conclusion:

   • The range of $f(x) = x - [[x]]$ is $[0, 1)$.

Thus, the range of $f(x) = x - [[x]]$ is $[0, 1)$.