Let $f:Y\rightarrow Z$ with $Y\subseteq X_I:=\Pi_{i\in I}\,X_i$. Then {\rm (a)}~$J\subseteq I$ is {\it essential} if there are $x,y\in Y$ such that $d(x,y)=J$ and $f(x)\neq f(y)$, where $d(x,y):=\{i\in I:x_i\neq y_i\}$; $J_f:=\{i:\{i\}$ is essential$\}$; an essential $J$ is {\it optimally essential} if no essential $J'\subseteq J$ satisfies $|J'|<|J|$; $\sJ\in\cJ_f$ if $\sJ$ is a maximal family of pairwise disjoint optimally essential sets; $\lambda_f:=\sup\{|\sJ|:\sJ\in\cJ_f\}$. {\rm (b)}~$f$ {\it depends on} $J\subseteq I$ if $[x,y\in Y, x_J=y_J]\Rightarrow f(x)=f(y)$; $\sD_f:=\{J\subseteq I:f$ depends on $J\}$; $\mu_f:=\min\{|J|:J\in\sD_f\}$. \underline{Theorem~1.}~$J\in\sD_f\Rightarrow\lambda_f\leq|J|$. \underline{Theorem~2.}~$\sJ\in\cJ_f\Rightarrow\bigcup\sJ\in\sD_f$. That context is strictly set-theoretic. Henceforth let $X_I$ and $Z$ be spaces with $Z$ Hausdorff, and let $f\in C(Y,Z)$. This is known: (*)~if $Y$ contains a $\sigma$-product then $J\in\sD_f$ iff $J_f\subseteq J$. The authors give examples to show: $J_f\in\sD_f$ in (*) can fail, if any one of the three hypotheses are omitted; $\sJ,\sJ'\in\cJ_f$, with $|\sJ|\neq|\sJ'|$, can occur; $J\in\sD_f\Rightarrow|J|>\lambda_f$ can occur; $\sJ\in\cJ_f\Rightarrow|\sJ|<\lambda_f$ can occur; $|J|\leq\lambda_f\Rightarrow J\notin\sD_f$ (hence, $\lambda_f<\mu_f$) can occur. The authors' interest in (*) is motivated by their observation that when $X_I$ has the $\kappa$-box topology, its obvious analogue, say (*)$_\kappa$, can fail. They propose and prove (what seem to be) appropriate modifications of (*)$_\kappa$.