Let [ x ] be the greatest integer x . Then the number of points in the interval ( - 2 , 1 ) , where the function f ( x ) = | [ x ] | + x - [ x ] is discontinuous, is _______ . [2023]

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Solution

Q.
Let $[x]$ be the greatest integer $\leq x$ . Then the number of points in the interval $(- 2, 1)$ , where the function $f (x) =$ $| [x] |$ $+ \sqrt{x - [x]}$ is discontinuous, is _______ . [2023]

Ans.
(2)

$We have, f (x) =$ $| [x] |$ $+ \sqrt{x - [x]}$

Where $[x]$ is G.I.F. discontinuous at $x \in I$ only. Then,

at $x = - 1, f (- 1^{+}) = 1 + 0 = 1$ and $f (- 1^{-}) = 2 + 1 = 3$

at $x = 0, f (0^{+}) = 0 + 0 = 0$ and $f (0^{-}) = 1 + 1 = 2$

Hence, $f (x)$ is discontinuous at two points.

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