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Topology

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Moderate

Function Applied To A Norm

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Let $X$ be a linear space of dimension at least 1, and $x\mapsto \|x\|$ a norm on $X$.

Suppose that a function $f:[0,\infty) \to \mathbb{R}\,$ is given, and define:

$$ N(x):=f(\|x\|), \ x\in X. $$

You are told that $N$ is a norm on $X$.

Choose ALL the correct statements from the list:

A

$f\,$ is a continuous function.

B

the inverse image $f^{-1}(\{0\})$ is an empty set.

C

the inverse image $f^{-1}(\{0\})$ consists of a single point.

D

$f\,$ is a restriction of a linear map from $\mathbb{R}$ to $\mathbb{R}$.

E

the inverse image $f^{-1}(\{0\})$ consists of at least two points.

F

$f\,$ is a strictly increasing function.