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JEE Main202228 Jun 2022Morning ShiftMathematicsContinuity and DifferentiabilityActual

Let f : ℝ → ℝ be defined as f x = e x , x < 0 a e x + x - 1 , 0 ≤ x < 1 b + sin π x , 1 ≤ x < 2 e - x - c , x ≥ 2 where a , b , c ∈ ℝ and t denotes greatest integer less than or equal to t . Then, which of the following statements is true?

Options

  1. AThere exists a , b , c ∈ ℝ such that f is continuous of ℝ .
  2. BIf f is discontinuous at exactly one point, then a + b + c = 1
  3. CIf f is discontinuous at exactly one point, then a + b + c ≠ 1 .
  4. Df is discontinuous at atleast two points, for any values of a , b and c .

Correct answer

C. If f is discontinuous at exactly one point, then a + b + c ≠ 1 .

Step-by-step solution

Given f x = e x , x < 0 a e x + x - 1 , 0 ≤ x < 1 b + sin π x , 1 ≤ x < 2 e - x - c , x ≥ 2 i.e. f x = 0 , x < 0 a e x - 1 , 0 ≤ x < 1 b , x = 1 b - 1 , 1 < x < 2 - c , x ≥ 2 Here, we know f x is discontinuous at x = 1 as b ≠ b - 1 For f x to be continuous at x = 0 ;   a = 1 For f x to be continuous at x = 2 ;   b + c = 1 Hence, a + b + c = 2

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