If f(x) is a continuous function ∀x∈R and f(x)∈(1,30),andg(x)=[f(x)a], where [.] denotes the greatest integer function), is continuous ∀x∈R then the least positive integral value of a is

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If $f (x)$ is a continuous function $\forall x \in R$ and $f (x) \in (1, \sqrt{30}), and g (x) = [\frac{f (x)}{a}],$ where $[.]$ denotes the greatest integer function), is continuous $\forall x \in R$ then the least positive integral value of a is

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Detailed Solution

Given g(x) is a continuous function ∀x∈R and the range of f(x) is ( $1, \sqrt{30}$ ) and g(x)=[ $\frac{f (x)}{a}$ ] is continuous ∀x∈R

Hence for g(x) to be continuous, $a > \sqrt{30}$

Because if we take any value of a between ( $1, \sqrt{30}$ ) like 3,4,5 then, when f(x) varies from 1 to $\sqrt{30}$ and due to greatest integer function g(x) become discontinuous at a=3,4,5.