Differential Calculus-1
Differential Calculus-1
Allen Star Batch
Grade 12
Question:
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: Absolute value compositions create piecewise-defined functions whose ranges can be identified by evaluating each interval separately.
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