Let f ( x ) = [ x 2 - x ] + | - x + [ x ] | , where x and [ t ] denotes the greatest integer less than or equal to t. Then, f is: [2023]

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Q.
Let $f (x) = [x^{2} - x] +$ $| - x + [x] |$ , where $x \in ℝ$ and $[t]$ denotes the greatest integer less than or equal to t. Then, $f$ is: [2023]

1 continuous at $x = 0$ , but not continuous at $x = 1$

2 continuous at $x = 0$ and $x = 1$

3 not continuous at $x = 0$ and $x = 1$

4 continuous at $x = 1$ , but not continuous at $x = 0$

Ans.
(4)

We have, $f (x) = [x^{2} - x] +$ $| - x + [x] |$

$= [x (x - 1)] +$ $| - x + x - {x} |$

$\Rightarrow f (x) = [x (x - 1)] + {x}$

$f (0^{+}) = - 1 + 0 = - 1$ $R.H.L. = f (1^{+}) = 0 + 0 = 0$

$f (0) = 0$ $f (1) = 0$

$L.H.L. = f (1^{-}) = - 1 + 1 = 0$

$∴$ $f (x) is continuous at x = 1, and discontinuous at x = 0 .$

Solution

Q.
Let $f (x) = [x^{2} - x] +$ $| - x + [x] |$ , where $x \in ℝ$ and $[t]$ denotes the greatest integer less than or equal to t. Then, $f$ is: [2023]

1 continuous at $x = 0$ , but not continuous at $x = 1$

2 continuous at $x = 0$ and $x = 1$

3 not continuous at $x = 0$ and $x = 1$

4 continuous at $x = 1$ , but not continuous at $x = 0$

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$f (0^{+}) = - 1 + 0 = - 1$	$R.H.L. = f (1^{+}) = 0 + 0 = 0$
$f (0) = 0$	$f (1) = 0$
	$L.H.L. = f (1^{-}) = - 1 + 1 = 0$

Solution

Q. Let f(x)=[x2-x]+|-x+[x]|, where x∈ℝ and [t] denotes the greatest integer less than or equal to t. Then, f is: [2023]

1 continuous at x=0, but not continuous at x=1

2 continuous at x=0 and x=1

3 not continuous at x=0 and x=1

4 continuous at x=1, but not continuous at x=0

Ans. (4) We have, f(x)=[x2-x]+|-x+[x]| =[x(x-1)]+|-x+x-{x}| ⇒f(x)=[x(x-1)]+{x} f(0+)=-1+0=-1 R.H.L.=f(1+)=0+0=0 f(0)=0 f(1)=0 L.H.L.=f(1-)=-1+1=0 ∴ f(x) is continuous at x=1, and discontinuous at x=0.

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Q.
Let $f (x) = [x^{2} - x] +$ $| - x + [x] |$ , where $x \in ℝ$ and $[t]$ denotes the greatest integer less than or equal to t. Then, $f$ is: [2023]

1 continuous at $x = 0$ , but not continuous at $x = 1$

2 continuous at $x = 0$ and $x = 1$

3 not continuous at $x = 0$ and $x = 1$

4 continuous at $x = 1$ , but not continuous at $x = 0$