Let [ t ] denote the greatest integer ≤ t . The number of points where the function f ( x ) = [ x ] x 2 - 1 + sin π [ x ] + 3 - [ x + 1 ] , x ∈ ( - 2 , 2 ) is not continuous is _____ .

Question

Let [ t ] denote the greatest integer &#8804; t . The number of points where the function f ( x ) = [ x ] x 2 - 1 + sin &#960; [ x ] + 3 - [ x + 1 ] , x &#8712; ( - 2 , 2 ) is not continuous is _____ .

Quantrex Academy · Accepted Answer

Correct answer: 0

As [ . ] is discontinuous at integers so we will check at x = - 1 , &#160; 0 , &#160; 1 only at x = - 1 [ - 1 - ] = - 2 and [ - 1 + ] = - 1 we have LHL = ( - 2 ) &#183; ( 0 ) + sin &#960; + 1 = 1 RHL = ( - 1 ) &#183; ( 0 ) + sin &#960; 2 - 0 = 1 f ( - 1 ) = 1 &#8658; Continuous at x = - 1 Again at x = 0 [ 0 - ] = - 1 &#160; &#38; &#160; [ 0 + ] = 0 LHL = ( - 1 ) &#183; ( 1 ) + sin &#960; 2 - 0 = 0 RHL = 0 + sin &#960; 3 - 1 &#8800; LHL Hence, discontinuous at x &#160; = &#160; 0 Again at x = 1 , [ 1 - ] = 0 &#160;

Let [ t ] denote the greatest integer ≤ t . The number of points where the function f ( x ) = [ x ] x 2 - 1 + sin π [ x ] + 3 - [ x + 1 ] , x ∈ ( - 2 , 2 ) is not continuous is _____ .

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Let [ t ] denote the greatest integer &#8804; t . The number of points where the function f ( x ) = [ x ] x 2 - 1 + sin &#960; [ x ] + 3 - [ x + 1 ] , x &#8712; ( - 2 , 2 ) is not continuous is _____ .

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Let [ t ] denote the greatest integer ≤ t . The number of points where the function f ( x ) = [ x ] x 2 - 1 + sin π [ x ] + 3 - [ x + 1 ] , x ∈ ( - 2 , 2 ) is not continuous is _____ .