Let $f(x) = x^2 \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$. Which of the following is true?

A. f(x) is not continuous at x = 0
B. f(x) is not differentiable at x = 0
C. f'(x) is not continuous at x = 0
D. f'(x) is continuous at x = 0

✅ Correct Answer: f'(x) is not continuous at x = 0

Explanation

$f(x)$ is continuous at $x = 0$ $f'(x) = 2x \sin(1/x) - \cos(1/x)$ for $x \ne 0$ At $x = 0$, $f'(0) = 0$ Limit of $f'(x)$ as $x \to 0$ does not exist Hence $f'(x)$ is not continuous at $x = 0$

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Let $f(x) = x^2 \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$. Which of the following is true? | ACME Academy