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 
$$|g(t)-f(t)|<d,\ \forall t\in [a,b].$$



Approximation of an injective continuous curve by injective piecewise linear curves
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](http://dx.doi.org/10.1112/blms/12.1.34).


