Part CCSIR NET December 2024shifting-is-tame-and-rescaling-is-not

Shifting is tame and rescaling is not

Let $f : \mathbb{R} \to \mathbb{R}$ be a uniformly continuous function. For each positive integer n and $x \in \mathbb{R}$ , let $g_{n}$ and $h_{n}$ be defined by $g_{n}(x) = f(x + 1/n)$ and $h_{n}(x) = f($ nx). Which of the following statements are necessarily true?

A. $(g_{n})_{n}\ge_{1}$ converges uniformly on any compact subset of $\mathbb{R}$ but not on $\mathbb{R}$ .
B. $(g_{n})_{n}\ge_{1}$ converges uniformly on $\mathbb{R}$ .✓ Correct
C.There exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on $\mathbb{R}$ .
D.For every compact subset $K \subseteq \mathbb{R}$ , there exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on K.

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 pointwise vs uniform.

See pricing

50 are analysed free — try those first.

The trap it tests

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

Drill statements like this

Related counterexample: fₙ → f uniformly ⇒ fₙ′ → f′

Shifting is tame and rescaling is not

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