Pointwise functions on vector space (reference request) I'm just looking for a reference or anything related to the question I have outlined below. I have personally never come across anything like this -

Suppose for $\mathbb{R}^n$ we define a continuous pointwise function as a function $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ that can be written in the form $$F(v) = (f_1(v_1), f_2(v_2), f_3(v_3),\cdots,f_n(v_n))$$

w.r.t some basis, and where the $f_i$'s are continuous functions from $\mathbb{R} \rightarrow \mathbb{R}$.

Now my question is if we consider the set of functions from $\mathbb{R}^n \rightarrow \mathbb{R}^n$ that are generated by composing these continuous pointwise functions, is that set of functions equal to, or dense in set of all continuous functions from $\mathbb{R^n} \rightarrow \mathbb{R^n}$?

Note that when composing two continuous pointwise functions, they do not have to be pointwise w.r.t the same basis.
Composing 2 of your continuous pointwise functions just gives another continuous  pointwise function . If $ F=(f_1,...,f_n) ,and G=(g_1,...g_n) then FoG=((f_1)o(g_1)...(f_n)o(g_n))$
Composing 2 of your continuous pointwise functions just gives another continuous  pointwise function . If $ F=(f_1,...,f_n) ,and G=(g_1,...g_n) then FoG=((f_1)o(g_1)...(f_n)o(g_n))$