There is a nice theorem characterizing continuous functions $A \to Y^X$ where $Y^X$ is equipped with product topology (pointwise convergence topology).
Is there a similar theorem for functions of the form $Y^X \to A$? I am most interested in a case where $Y^X = 2^\mathbb{N}$ and $A = \mathbb{R}$ (with the standard topology). Could someone give me some intuitions how continuous functions in this space (that is $A^{(Y^X)}$) look like (or at least some references)?
Thanks in advance!
Edit:
There are non-constant continuous functions in this space, e.g. let $f(A) = \sum_{k \in A} \frac{1}{2^{k}},$ then $\lim_n f(a_n) = f(\lim_n a_n)$ for every $a_n$ convergent in $2^\mathbb{N}$ (in product topology = pointwise convergence topology, that means every sub-coordinate is constant from some point). Take any $\varepsilon > 0$, then there is a finite number $k_0$ such that $\sum_{i > k_0}2^{-i} < \frac{1}{2}\varepsilon$, so there is $M_0$ such that $\forall k \leq k_0.\ \left(\forall m > M_0.\ k \in a_m\right) \lor \left(\forall m > M_0.\ k \notin a_m\right),$ and thus for all $m > M_0$ we have $|f(a_m) - f(a)| < \varepsilon$.