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

Differentiability costs one more power than continuity

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

A.f is differentiable at 0 when $α > 1$ .✓
B.f is differentiable at 0 when $α = 1/2$ .
C.f is continuous at 0 when $α = 0$ .
D.f is continuous at 0 when $α > 0$ .✓

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.

Differentiability costs one more power than continuity

More on this topic