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Topology

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The Euclidean norm applied to a function may generate a metric

TOPO-WV1BB4

Let $f:\mathbb{R}\to \mathbb{R}\,$ and define $f:\mathbb{R} \times \mathbb{R}\to \mathbb{R}$ by:

$$ d(x,y)=|f(x)-f(y)|, \ x,y\in \mathbb{R}$$

You are told that $\,d\,$ is a metric.

What is necessarily TRUE about $f\,$?

A

$f\,$ is onto,

B

$f\,$ is non-negative,

C

$f\,$ is bijective,

D

$f\,$ is one-to-one,

E

$f\,$ is strictly monotone (increasing or decreasing),

F

$f\,$ is continuos at $0$.