Approximation of continuous functions by \(\alpha \)-dense curves

Given a compact and densifiable metric space D, and a continuous function \(f:D\longrightarrow \mathbb {R}^{m}\) $f:D⟶{\mathbb{R}}^m$ , we present a method to approximate f(D) by \(\Gamma _{n}([0,1])\) ${\mathrm{\Gamma }}_n([0,1])$ , where \(\Gamma _{n}:[0,1]\longrightarrow \mathbb {R}^{m}\) ${\mathrm{\Gamma }}_n:[0,1]⟶{\mathbb{R}}^m$ is certain single variable function. To be more precise, given \(\varepsilon >0\) $\epsilon >0$ , we can construct a function \(\Gamma _{n}\) ${\mathrm{\Gamma }}_n$ such that \(d_{\textrm{HP}}\big ( f(D), \Gamma _{n}([0,1]) \big ) \le \varepsilon \) $d_{\text{HP}}(f(D),{\mathrm{\Gamma }}_n([0,1]))\le \epsilon$ , where \(d_{\textrm{HP}}\) $d_{\text{HP}}$ is the Hausdorff–Pompeiu metric. Our main tool are the so called \(\alpha \) $\alpha$ -dense curves, which allow us to transform a multidimensional problem into a single variable one. Some numerical examples are provided as well as a brief discussion about the proposed method.