Let denote the greatest integer less than or equal to Let be a function defined by Let be the set of all points in the interval [0, 8] at which is not continuous. Then is equal to

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Solution

Q.
Let $[t]$ denote the greatest integer less than or equal to $t .$ Let $f : [0, \infty) \to R$ be a function defined by $f (x) = [\frac{x}{2} + 3] - [\sqrt{x}] .$ Let $S$ be the set of all points in the interval [0, 8] at which $f$ is not continuous. Then $\sum_{a \in S} a$ is equal to ______ . [2024]

Ans.
(17)

Given, $f (x) = [\frac{x}{2} + 3] - [\sqrt{x}] = [\frac{x}{2}] - [\sqrt{x}] + 3$

$[\frac{x}{2}]$ $is discontinuous at 2,4, 6, 8 and [\sqrt{x}] discontinuous at 1, 4$

$∴ f (x) is discontinuous at 1, 2, 6, 8$

$[∵ f (4^{+}) = 3 = f (4^{-})]$

$∴ \sum_{a \in S} a = 1 + 2 + 6 + 8 = 17$

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