A characterization of the set of all integers by a composite functional equation

Let $$f: \mathbb {R} \rightarrow \mathbb {R}$$ . The Heaviside function H is defined by $$ H(x) = {\left\{ \begin{array}{ll} 1 \quad (x > 0)\\ 0 \quad (x \le 0). \end{array}\right. } $$ In this paper, we study the composite functional equation $$\begin{aligned} f(x + f(y)) = (x + f(y)) G_f (x), \end{aligned}$$ where $$\begin{aligned} G_f (x) = f(H(x) + f(x) - x) + f(H(1 - x) + f(1 - x) - (1 - x)). \end{aligned}$$ Define the function $$I: \mathbb {R} \rightarrow \mathbb {R}$$ by setting $$ I(x) = {\left\{ \begin{array}{ll} x \quad (x \in \mathbb {Z})\\ 0 \quad (x \in \mathbb {R} \setminus \mathbb {Z}). \end{array}\right. } $$ We prove that $$f = I$$ holds in the case $$f(1) \ne 0$$ . We can regard this fact as a characterization of $$\mathbb {Z}.$$