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?

Question

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?, There exists a , b , c ∈ ℝ such that f is continuous of ℝ ., If f is discontinuous at exactly one point, then a + b + c = 1, If f is discontinuous at exactly one point, then a + b + c ≠ 1 ., f is discontinuous at atleast two points, for any values of a , b and c .

MARKS App · Accepted Answer

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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