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JEE Advanced2026MathematicsContinuity and DifferentiabilityActual

Let R denote the set of all real numbers. Let f : R R be an arbitrary function and let g : R R be the function defined by g(x) = x f(x) , for all x R . Then which of the following statements is (are) TRUE?

Options

  1. AThe function g is always continuous at x = 0
  2. BIf f is continuous at x = 0 , then g is differentiable at x = 0
  3. CIf g is differentiable at x = 0 , then f is continuous at x = 0
  4. DIf g is differentiable at x = 0 , then _ x 0 f(x) exists

Correct answer

B. If f is continuous at x = 0 , then g is differentiable at x = 0

Step-by-step solution

Given g(x) = x f(x) for all x R . For option A: Let f(x) = 1 x for x 0 and f(0) = 0 . Then g(x) = 1 for x 0 and g(0) = 0 . _ x 0 g(x) = 1 g(0) . Thus, g is not necessarily continuous at x = 0 . Option A is false. For option B: The derivative of g at x = 0 is given by: g'(0) = _ x 0 g(x) - g(0) x - 0 = _ x 0 x f(x) - 0 x = _ x 0 f(x) . If f is continuous at x = 0 , then _ x 0 f(x) = f(0) . Since f(0) is a finite real number, g'(0) exists and g is differentiable at x = 0 . Option B is true. For option C: Let f(x) = 0

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