Let $A = \{a_1,\dots,a_m\}$ be a finite set endowed with a discrete topology and let $X = A^{\Bbb N}$ be the product topological space. I wonder which bounded functions $f:X\to\Bbb R$ are continuous on $X$.
For example, it is clear that if $f$ depends only on a finite number of coordinates then $f\in C(X)$, i.e. if there exists some finite $n$ such that $ f(x_1,\dots,x_n,x_{n+1},x_{n+2},\dots) = f(x_1,\dots,x_n,x'_{n+1},x'_{n+2},\dots) \quad \forall x_{n+1},x_{n+2},x'_{n+1},x'_{n+2},\dots $ then $f\in C(X)$. Thus it seems that only dependents on infinitely many coordinates may violate continuity. I would be happy, if one could tell me whether there are some useful necessary/sufficient conditions to assure $f\in C(X)$.
In particular, if $B\subset A$ and $1_B(a)$ is the indicator (characteristic) function of $B$, does it hold that $ g(x):=\limsup\limits_{k\to\infty}1_B(x_k) $ is a continuous function on $X$.