JEE Advanced 2026 Mathematics question

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) = xf(x), for all x .

Then which of the following statements is (are) TRUE ?

• (A)The function g is always continuous at x = 0
• (B)If f is continuous at x = 0 , then g is differentiable at x = 0Correct
• (C)If g is differentiable at x = 0 , then f is continuous at x = 0
• (D)If g is differentiable at x = 0 , then existsCorrect