Limits and Continuity: JEE Advanced MCQ

Solve this quality-checked JEE multiple-choice question, then review the correct answer and explanation.

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JEE AdvancedHardQuestion 240017

Question

If f(x) = x² sin(1/x) for x ≠ 0 and f(0) = 0, then f is:

A
Continuous at 0 but not differentiable
B
Both continuous and differentiable at 0
Correct
C
Neither continuous nor differentiable
D
Differentiable but not continuous

Correct answer

Both continuous and differentiable at 0

Explanation

Continuity: |f(x)| = |x² sin(1/x)| ≤ x² → 0 as x → 0 = f(0) ✓. Differentiability: f'(0) = lim(x→0) f(x)/x = lim x sin(1/x). Since |x sin(1/x)| ≤ |x| → 0, f'(0) = 0 ✓.

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