Let $\text{Cont}(\mathbb{R},\mathbb{R})$ denote the set of continuous self-maps of $\mathbb{R}$ and let $\mathbb{R}^\mathbb{R}$ denote the set of all self-maps of $\mathbb{R}$, endowed with the product topology. Is $\text{Cont}(\mathbb{R},\mathbb{R})$ dense in $\mathbb{R}^\mathbb{R}$?



Is $\text{Cont}(\mathbb{R},\mathbb{R})$ dense in $\mathbb{R}^\mathbb{R}$?
Yes.


Let $g: \mathbb{R} \to \mathbb{R}$ be an abritrary function.


Let $\mathcal{F}$ denote the set of all finite subsets of $\mathbb{R}$. We endow $\mathcal{F}$ with the order $\subseteq$, which renders it a directed set.


For each $F \in \mathcal{F}$, choose a continuous function $f\_F$ which fulfils $f\_F(x) = g(x)$ for all $x \in F$. Then the net $(f\_F)\_{F \in \mathcal{F}}$ converges pointwise to $g$.


