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BITSAT2017MathematicsFunctionsActual

If a function f : R → R satisfy the equation f ( x + y ) = f ( x ) + f ( y ) , ∀ x , y and the function f ( x ) is continuous at x = 0 , then

Options

  1. Af ( x ) is continuous for all positive real values of x
  2. Bf ( x ) is continuous for all x
  3. Cf ( x ) = 0 for all x
  4. DNone of the above

Correct answer

B. f ( x ) is continuous for all x

Step-by-step solution

∵   lim x → 0 f x = f 0 Let a be any point Now, at x = a , lim x → a f x = lim h → 0 f a + h = lim h → 0 f a + lim h → 0 f h = f a + f 0 = f a + 0 = f a ∴   f x is continuous at x = a , where a is any arbitrary point. Hence, f x is continuous for all x .

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