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?

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Correct answer: D. If g is differentiable at x = 0, then _ x 0 f(x) exists

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 for x 0 and f(0) = 1. Then g(x) = 0 for all x R . Here, g is differentiable at x = 0 with g'(0) = 0. However, _ x 0 f(x) = 0 f(0), so f is not continuous at x = 0. Option C is false. For option D: If g is differentiable at x = 0, then g'(0) exists. Since g'(0) = _ x 0 g(x) - g(0) x = _ x 0 f(x), the limit _ x 0 f(x) must exist. Option D is true. Answer: If f is continuous at x = 0, then g is differentiable at x = 0; If g is differentiable at x = 0, then _ x 0 f(x) exists

JEE Advanced Mathematics Question (2026) — Solution

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