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JEE Main202311 Apr 2023Morning ShiftMathematicsContinuity and DifferentiabilityActual

Let f x = x 2 - x + - x + x , where x ∈ ℝ and t denotes the greatest integer less than or equal to t . Then, f is

Options

  1. Acontinuous at x = 0 , but not continuous at x = 1
  2. Bcontinuous at x = 1 , but not continuous at x = 0
  3. Ccontinuous at x = 0 and x = 1
  4. Dnot continuous at x = 0 and x = 1

Correct answer

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

Step-by-step solution

Given, f x = x 2 - x + - x + x ⇒ f x = x 2 - x + x - x ,     as   - A = A ⇒ f x = x 2 - x + x ,   as   x + x = x ⇒ f x = x 2 - x + x    ( ∵ x ≥ 0 ) Now at x = 0 , f 0 = 0 And f 0 + = - 1 ,   as   x 2 - x < 0   for   x → 0 + So, function is discontinuous at x = 0 Now at x = 1 ,   f 1 = 0 , f 1 + = 0 + 0 = 0 and f 1 - = - 1 + 1 = 0 Hence, the function is continuous at x = 1 .

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