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JEE Main202628 January 2026Evening ShiftMathematicsContinuity and DifferentiabilityActual

Let f(x)= _ 0 ( x-x^ ( 2 ) (x-1) 1+x^ ( 2 ) (x-1) ), x R . Consider the following two statements : (I) f(x) is discontinous at x=1 . (II) f(x) is continous at x=-1 . Then,

Options

  1. AOnly (II) is True
  2. BNeither (I) nor (II) is True
  3. CBoth (I) and (II) are True
  4. DOnly (I) is True

Correct answer

B. Neither (I) nor (II) is True

Step-by-step solution

Taking the limit as 0 , we note x^ (2/ ) 0 for |x| 1 . This gives: f(x) = cases x & x 1^- - (x-1) (x-1) & x 1^+ cases Continuity at x = 1 : RHL = _ x 1^+ - (x-1) (x-1) = -1 LHL = _ x 1^- x = -1 , f(1) = -1 f(x) is continuous at x = 1 . So Statement (I) is false. Continuity at x = -1 : f(x) = cases - (x-1) -(x-1) & x -1^- x & x -1^+ cases RHL = _ x -1^+ x = -1 LHL = _ x -1^- - (x-1) -(x-1) = 2 -2 Since LHL RHL, f(x) is discontinuous at x = -1 . So Statement (II) is also false. Hence, neither (I) nor (II) is true.

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