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How to see if the set $B=\{(x,\frac1x), x>0\}$ is connected on $(\mathbb{R},|.|)$

Thank you

Vrouvrou
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    You can see it by drawing the graph of $y = 1/x$ for $x>0$. As for proving it: http://math.stackexchange.com/questions/2018848/proof-that-a-continuous-function-maps-connected-sets-into-connected-sets?noredirect=1&lq=1 – Winther Feb 14 '17 at 17:49

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$B=H(0,+\infty)$, where $H$ is the continuous function $$H:(0,+\infty) \to (0,+\infty) \times (0,+\infty), \ H(x)=\left(x,\frac1x\right)$$ and $(0,+\infty)$ is connected. Recall that the image of a connected set by a continuous function is connected.

A. Salguero-Alarcón
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Hint

Pick $(a,b), (c,d) \in B$ and consider $\varphi \colon [0,1] \to A$ $$ \varphi(t)= \left(a+(c-a)t, \; \frac{1}{a+(c-a)t}\right).$$ Now argue that $\varphi$ is continuous.