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BITSAT2015MathematicsContinuity and DifferentiabilityActual

Let f: R R be a function such that f(x+y)=f(x)+f(y), x, y R If f ( x ) is differentiable at x =0 , then which one of the following is incorrect?

Options

  1. Af ( x ) is continuous, x R
  2. Bf ^ ( x ) is constant, x R
  3. Cf ( x ) is differentiable, x R
  4. Df ( x ) is differentiable only in a finite interval containing zera.

Correct answer

D. f ( x ) is differentiable only in a finite interval containing zera.

Step-by-step solution

Let f(x+y)=f(x)+f(y), x, y R Put x=0=y f(0)=f(0)+f(0) f(0)=0 Now, f^ (0)= _ h 0 f(0+h)-f(0) h f^ (0)= _ h 0 f(h) h Now, f(x)= _ h 0 f(x+h)-f(x) h = _ h 0 f(x)+f(h)-f(x) h array l f ^ ( x )= _ h 0 f ( h ) h = f ^ (0) f ( x )= x f (0)+ C array But f(0)=0 C =0 Hence, f(x)=x f(0), x R Clearly, f ( x ) is everywhere continuous and differentiable and f ^ ( x ) is constant. x R

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