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If f x = x 2 : w h e n x i s r a t i o n a l 2 - x : w h e n x i s i r r a t i o n a l , then

Options

  1. Af x is continuous for all real x
  2. Bf x is discontinuous for all real x
  3. Cf x is continuous only at x = 1 , - 2
  4. Df x is discontinuous only at x = 1 , - 2

Correct answer

C. f x is continuous only at x = 1 , - 2

Step-by-step solution

Let a be any real number. lim x → a ⁡ f x = lim x → a ⁡ x = a 2 (when x → a through rational values) lim x → a ⁡ f x = lim x → a ⁡ 2 - x = 2 - a (when x → a through irrational values) lim x → a ⁡ f x will exist only when a 2 = 2 - a i.e., when a = 1 , - 2 Thus, if a ≠ 1 or - 2 then lim x → a ⁡ f x will not exist Hence, f x will be discontinuous at x = a where a ≠ 1 or - 2

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