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How do I show that the set $A = \{(x,y) \in R^2: x \geq 0, y \geq 0\} \cup \{(x,y) \in R^2: x \leq 0, y \leq 0\}$ is path connected. I know that I need to construct a continuous function $f:[0,1] \rightarrow A$ such that $f(0) = x, f(1) = y$ for any pair of points $x, y \in A$ but I don't know how to construct this function.

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You take the segment from $x$ to $(0,0)$, and you paste this with the segment from $(0,0)$ to $y$. It is continuous, using the pasting lemma and being the concatenation of two continuous paths. Writing the exact expression is not diffcult, it is simple vector calculus.

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