Given a general function $f:\mathbb{R}\rightarrow \mathbb{R}$ we might say it has type $\mathbb{R}^\mathbb{R}$.
But if it can be written as a Taylor series:
$f(x)=\sum\limits_{n=0}^\infty a_n x^n$
Then $f$ is completely determined by the set $\{a_n\}$. which is of the form $a:\mathbb{N}\rightarrow\mathbb{R}$.
And so we have two different types which represent the same function $\mathbb{R}^\mathbb{R}$ and $\mathbb{R}^\mathbb{N}$.
Should we say that $\mathbb{R}^\mathbb{N} \subset \mathbb{R}^\mathbb{R}$? And that $\mathbb{R}^\mathbb{N}$ is the type of members of a set which contains all functions that can be written as Taylor series? What is the correspondence?
I think this would exclude functions like $1/\sin(x)$ which have an infinite number of poles. (If we let $\infty$ be a member of $\mathbb{R}$.)
This is confusing because it seems to suggest that we can have $f:\mathbb{R}\rightarrow \mathbb{R}$ where $f\in \mathbb{R}^\mathbb{N}$ ??