JEE Main 202312 AprilMorning Shift

Exam: JEE Main
Year: 2023
Date: 12 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2023 · 12 April · Morning Shift · Mathematics · Continuity and Differentiability

Let [x] be the greatest integer ≤ x . Then the number of points in...

Let $\left[x\right]$ be the greatest integer $\le x$ . Then the number of points in the interval $( – 2, 1 )$ where the function $f$ \left(x\right) = $\left|\left[x\right]\right| + \sqrt{x - \left[x\right]}$ is discontinuous, is _____.

Correct answer

Step-by-step solution

Given,

$f\left(x\right) = \left|\left[x\right]\right| + \sqrt{x - \left[x\right]}$

$x - \left[x\right] \ge 0 ⇒ x \in R$

Now using the property of greatest integer function and simplifying the function we get,

$f\left(x\right) = \left\{$ $\begin{array}{ll}2 + \sqrt{x + 2}, - 2 < x < - 1 \\ 1 + \sqrt{x + 1}, - 1 \le x < 0 \\ \sqrt{x}, 0 \le x < 1\end{array}$ $\right)Now plotting the diagram we get,<img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/b6c78f72-2e3a-4af6-9cb4-16130816aa9e.png" style="width: 300px; height: 223px;" />Now from the diagram we can say that,$ f $\left(x\right)$ is discontinuous at two points $x = \left\{– 1, 0\right\}$ .

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