I need to determine whether the following is an equivalence relation on $X$:
$X = \{ f|f: A \to \mathbb{R}, A \subseteq \mathbb{R}\}$ (the set of functions from a subset of $\mathbb{R}$ to $\mathbb{R}$), functions $f: A \to \mathbb{R}$ and $g: B \to \mathbb{R}$ "partially agree" if and only if their restrictions on $A \cap B$ are equal:
$f\sim g$ if and only if $f|_{A \cap B} = g|_{A \cap B}$.
I assume that showing reflexivity and symmetry are obvious. Showing transitivity is the hard part.
I have a hunch that the relation is not reflexive, but I'm having some trouble wrapping my head around how to show that if $f|_{A\cap B} = g|_{A \cap B}$ and $g|_{B \cap C} = h|_{B \cap C}$, then $f|_{A \cap C} = h|_{A \cap C}$ (at least this is what I think I'm supposed to show. All these sets, as well as the whole idea of a function restriction are confusing me a little.