Continuity
Continuity of GIF
MMTS_Full_Test_10
Grade 12
Question:
If $f(x)$ is a continuous function $\forall x\in\mathbb{R}$ and the range of $f(x)$ is $(2,\sqrt{26})$ and $g(x)=\left[\dfrac{f(x)}{C}\right]$ is continuous $\forall x\in\mathbb{R}$, then the least positive integral value of $C$ is (where $[\cdot]$ denotes GIF)
Step-by-Step Solution
Key Concept: $g(x)$ continuous means $f(x)/C$ never hits an integer
Range of $f$ is $(2,\sqrt{26})\approx(2,5.099)$. For $g$ continuous, $f(x)/C$ in $(2/C,\sqrt{26}/C)$ must contain no integer. Least $C=6$: $(1/3, 0.85)$ contains no integer.
Correct Answer: 3