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Topology

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Moderate

More About Functions And Metrics

TOPO-NV715E

Consider functions $f,g:\mathbb{R}\to\mathbb{R}$ defined by:

$$ f(x)=\left\{ \begin{array}{ll} 10 x, & x\geq 0\\\ x, & x<0 \end{array} \right. \ \mbox{ and } \ g(x)=\left\{ \begin{array}{ll} x, & x\geq 0\\\ 10 x, & x<0 \end{array} \right. $$

Define respectively

$$ d_1(x,y) :=|f(x)-f(y)| \ \mbox{ and } \ d_2(x,y) :=|g(x)-g(y)|. $$

Which functions are metrics on $\mathbb{R}$?

A

$d_1$

B

$d_2$

C

$d_1+d_2$

D

$\min\{d_1,d_2\}$

E

$\max\{d_1,d_2\}$