Determine all functions f : $\mathbb{R}\to\mathbb{R}$ such that the equality $f([x]y)=f(x)[f(y)].\quad(1)$ holds for all $x,y\in\mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. Please provide the sum of all possible integer values of the constant C.

Let's think step by step and output the final answer within \boxed{}.
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