$g: \{0,...,9\}^{\Bbb{R}}\mapsto ?$ where $g = \left(\frac{1}{10^x}\right)_{x\in \Bbb{R_{\geq 0}}} \cdot$. Let $s \in \{0,...,9\}^{\Bbb{R}}$. Then $$g(s)= \left(\frac{1}{10^x}\right)_{x\in \Bbb{R_{\geq 0}}} \cdot s = \int_{0}^{\infty}\frac{s(x)}{10^x}dx$$

We can see that for all s such that $\lim_{x\to \infty}s(x)\neq 0$, the integral diverges. Therefore while the domain of $g$, the subset of integrable functions $X \subset\{0,...,9\}^{\Bbb{R}}$, has cardinality $\aleph_2$, its image is only $\Bbb{R_{\geq 0}}$. This means, compared to the countably infinite decimal expansion, the continuum…