Is there a discussion in literature of "families of functions with common restrictions"? What I mean is: given a family $F$ of functions we define $$ D_F = \left\{x \in \bigcup\nolimits_{f \in F}\operatorname{dom}(f)\ \middle|\ f_1(x) = f_2(x) \text{ for all } f_1, f_2 \in F\right\}. $$

It can be seen that for any $R \subseteq D_F$, $f_1|_R = f_2|_R$ (function restriction) for all $f_1, f_2 \in F$.

Is there already a study about the properties of $D_F$? In particular, is there some established notation for $D_F$ as a function of $F$?

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