Can a continuous real-valued function on a large product space depend on uncountably many coordinates?

Bockstein's theorem

Bockstein, M., Un théorème de séparabilité pour les produits topologiques, Fundam. Math. 35, 242-246 (1948). ZBL0032.19103.

This is the case of a product $\prod_{t \in T} X_t$ where all factors are second-countable. I that case any continuous function $\prod_{t \in T} X_t \to \mathbb R$ depends on countably many coordinates.

PLUG... See Theorem 2.1 in

Edgar, G. A., Measurability in a Banach space, Indiana Univ. Math. J. 26, 663-677 (1977). ZBL0361.46017.

where the special case $X = \mathbb R$ is done. That is, a continuous function $\mathbb R^T \to \mathbb R$ depends on only countably many coordinates.

Let $X$ be an uncountable discrete space with a distinguished element $0$. We view the product space $X^X$ as the space of maps $\phi: X \to X$. The set $$ E := \{ \phi \in X^X: \phi(\phi(0)) = 0 \}$$ is easily seen to be clopen, hence the indicator function $1_E: X \to {\bf R}$ is continuous, but depends on all of the (uncountably many) coordinates of $X^X$.

The key point here (which was inspired by Nate's comment based on the earlier incorrect attempt at solving this problem) is that deciding whether a given map $\phi$ belongs to $E$ requires only a finite number of (adaptive) evaluations of $\phi$, but the set of (non-adaptive) locations where $\phi$ could potentially need to be evaluated is uncountable.

Note that a similar construction works for $X \times \{0,1\}^X$ using the set $E := \{ (x, \phi) \in X \times \{0,1\}^X: \phi(x)=0\}$; thus even a single highly non-compact factor is enough to generate a counterexample. (But I am not sure what happens if one insists that all of the factors be sigma-compact, in particular can one construct a continuous function $f: {\bf N}^{\bf R} \to {\bf R}$ that depends on uncountably many coordinates?)