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

Differentiable does not mean continuously so

Consider the function f:R→Rf : \mathbb{R} \to \mathbb{R} defined by f(x)=x2sin⁡(1/x)f(x) = x^{2} \sin(1/x) if x ≠ 0, and f(x) = 0 if x = 0. Which of the following statements are true?

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

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, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Differentiable ⇒ continuously differentiable

More on this topic

The chapter behind this: Differentiation — MVT, Darboux, Taylor and the classic spoilers — free to read

From Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital

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