Part CCSIR NET December 2024differentiability-costs-one-more-power-than-continuity

Differentiability costs one more power than continuity

Let ‖·‖ denote the Euclidean norm on $\mathbb{R}^{2}$ and $\alpha \ge 0$ . Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a function such that |f(x)| ≤ ‖x‖ $^\alpha$ . Which of the following statements are necessarily true?

A.f is differentiable at 0 when $\alpha > 1$ .✓ Correct
B.f is differentiable at 0 when $\alpha = 1/2$ .
C.f is continuous at 0 when $\alpha = 0$ .
D.f is continuous at 0 when $\alpha > 0$ .✓ 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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Differentiability costs one more power than continuity

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