EDIT: The following answer was not correct, as pointed out in the comments (though the false part is now deleted). At least it shows that there cannot be a finite collection of functions with the demanded property, but the infinite case is still open:
$\textbf{Claim}$: Let $A \subset \mathbb{R}$ of continuum size and let $f: A \to \mathbb{R}$ then there exist $\mathcal{c}$ (contiuum size) many subsets of $A$ of size $\mathcal{c}$ such that for each such $X$, $f(X) \neq \mathbb{R}$.
$\textit{Proof}$: Consider a partition $P$ of $A$ of size $\mathcal{c}$ such that each element of $P$ has size $\mathcal{c}$. If for $\mathcal{c}$-many elements of $P$ $f(X) \neq \mathbb{R}$ then we are finished thus we may assume w.l.o.g that for each $X \in P$ $f(X) = \mathbb{R}$. Then we can pick for each $r \in A$ and each $X_i \in P$ an $x_i \in X_i$ such that $f(x_i) = r$. Thus we obtain for each $r \in$ $A$ a set $Y_r \subset \mathbb{R}$ of size $\mathcal{c}$ such that $f(Y_r) \neq \mathbb{R}$.
Now let $f_1, f_2,..$ be an arbitrary sequence of functions from $\mathbb{R}$ to $\mathbb{R}$. By our claim there is a set $A_1$ of continuum size such that $f(A_1) \neq \mathbb{R}$. Then using our claim again there exists an $A_2 \subset A_1$ such that $f_2(A_2) \neq \mathbb{R}$, and an $A_3 \subset A_2$ and so on.
Thus if $f_1,...,f_n$ are real valued functions, then there is a set $A$ of continuum size such that $f_i (A) \neq \mathbb{R}$ for each $i \le n$.