Part CCSIR NET December 2024differentiable-does-not-mean-continuously-so

Differentiable does not mean continuously so

Consider the function $f : R \to R$ defined by $f (x) = x^{2} sin (1/ x)$ if x ≠ 0, and f(x) = 0 if x = 0. Which of the following statements are true?

A.lim(x→0) f(x) exists.✓
B.f is continuous at 0.✓
C.f is differentiable at 0.✓
D.lim(x→0) f′(x) does not exist.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Differentiable ⇒ continuously differentiable

Differentiable does not mean continuously so

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