Is the space of $\mathbb{R}\to\mathbb{R}$ more huge than the space of all discrete functions? Assume that we have some general discrete function ${0,1,2,....n}\to\mathbb{R}$.
For each number I have a real value.
Let's infinite increase number $n$ (which is integer value), to approximate $\mathbb{R}\to\mathbb{R}$ function.
As I understand it is only approximation even in limit, isn't it?
Even if  I cover all integer values in the limit is cool, but we still have that $\mathbb{N}\ne\mathbb{R}$
(Cantor's Diagonal argument )
So the space of $\mathbb{R}\to\mathbb{R}$ is more huge then the space of all discrete functions? Isn't it?
(p.s. my background: I'm not familiar with functional-analysis)
 A: If you have a function $f\colon\{0,1,2,\dots,n\}\to\mathbb{R}$, you can consider the pair $(n,\hat{f})$, where $\hat{f}\colon\mathbb{N}\to\mathbb{R}$ is defined by
$$
\hat{f}(k)=
\begin{cases}
f(k) & \text{if $0\le k\le n$} \\[4px]
0 & \text{if $k>n$}
\end{cases}
$$
If $\mathscr{D}$ is the set of “discrete functions”, you get an injective map $\mathscr{D}\to \mathbb{N}\times\mathbb{R}^{\mathbb{N}}$ with $f\mapsto(n,\hat{f})$. Therefore
$$
|\mathscr{D}|\le|\mathbb{N}\times\mathbb{R}^{\mathbb{N}}|=
\aleph_0\cdot (2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}
$$
On the other hand,
$$
|\mathbb{R}^{\mathbb{R}}|=(2^{\aleph_0})^{2^{\aleph_0}}=2^{2^{\aleph_0}}
$$
and, by Cantor’s theorem,
$$
2^{\aleph_0}<2^{2^{\aleph_0}}
$$
A: This is not really an answer, but a suggestion on what you should actually be asking. I hope it's still helpful.
If I understand your question correctly, you are asking whether the set of functions $f: \mathbb{R} \to \mathbb{R}$ is larger in cardinality than the set of functions $g : \mathbb{N} \to \mathbb{R}$ (or the set of functions $h : A \to \mathbb{R}$, with $A$ a finite set, I'm not exactly sure by reading the question).
A function $f : A \to B$ is an element of the following Cartesian product: $$f \in \prod \limits_{a \in A} B = B^A.$$
Thus, what you want to look at is the cardinality of such product. I'm sorry but I don't have any idea what those cardinalities are: You need to ask someone more knowledgeable in set theory.
The point of my answer is that you should forget about those limit things, since this is fundamentally a set theoretic question. Specifically, a question on the cardinality of certain products.
