Is $f$ constant?

Question

Let $$ A = \prod_{i \in I} A_i$$ be a set, and let $f:A \to {\{0,1\}}$ such that $f$ is constant on every coordinate, i.e, if $(a_i)$ and $(b_i)$ differ for any one $j \in I$, then $f((a_i)) = f((b_i))$. Is $f$ constant?

I can see that $f$ is indeed constant while varying any finite number of coordinates, but, is it possible for $f$ to take different values on tuples that differ on $\textit{infinitely many}$ coordinates? Does this rely on the axiom of choice?

Thanks.

Answer 1

Hint : Take each $A_i$ equal to $\lbrace a,b \rbrace$, and let $f$ be the function that says if there are infinitely many occurrences of $a$. (i.e. it is $1$ is there are infinitely many, and $0$ if there aren't).

(I assume that $I$ is infinite here, but the case $I$ finite is easily dealt with).