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Given a family of continuous real functions $\mathcal{G}$, let$R_\mathcal{G}$ be a binary relation defined as follows: a continuous function$f\colon\mathbb{R}\to\mathbb{R}$ is in the relation with a closed set$E\subseteq\mathbb{R}$ if and only if there exists $g\in\mathcal{G}$ such that$f\upharpoonright E = g\upharpoonright E$. We consider a Galois connectionbetween families of continuous functions and hereditary families of closed setsof reals naturally associated to $R_\mathcal{G}$. We study complete latticesdetermined by this connection and prove several results showing the dependenceof the properties of these lattices on the properties of $\mathcal{G}$. In somespecial cases we obtain exact description of these lattices.

A Galois connection related to restrictions of continuous real functions