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Let $f:[a,b]\longrightarrow\mathbb{R}^2$ be an injective continuous function. For any $d>0$, does there exist a piecewise linear curve: $g:[a,b]\longrightarrow\mathbb{R}^2$ such that $g$ is also injective and $$|g(t)-f(t)|<d,\ \forall t\in [a,b].$$

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Yes, it can. This is essentially no different from a classical result that any Jordan curve can be approximated by a Jordan polygon; see, for example, Lemma 2 here.

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  • $\begingroup$ Mateusz Kwaśnicki: Thank you very much! $\endgroup$
    – user173856
    Commented Oct 4, 2017 at 10:22

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