SCHWEITZER 1997 Problem 7

Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality. $$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| (x, y)\in G^2$$ Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$ .