The function, $f(x) = [x] - [[x]]$, where $[ ]$ denotes greatest integer function:
is continuous for all positive integers
is discontinuous for all non-positive integers
has finite number of elements in its range
is such that its graph does not lie above the x-axis
Step-by-Step Solution
Key Concept: For any real x, if [x] = n (integer), then [[x]] = [n] = n, so f(x) = [x] - [[x]] = n - n = 0 for all x. The function is identically zero everywhere, which can be verified by recognizing that the greatest integer function applied to an integer returns that integer.
Evaluate the piecewise function $||x|| - |x||$ on each interval: at $x = -1$ the value is $0$; for $-1 < x < 0$ the value is $-1$; for $0 \leq x \leq 1$ the value is $0$; and for $1 < x \leq 2$ the value is $0$. The graph shows isolated points at $(-2, 0)$, $(-1, 0)$ and a filled region at $(-1, -1)$ with points at $(0, 0)$ and $(1, 0)$. The range of the function is $\{0, -1\}$.
Correct Answer: 1,2,3,4