2018-20 Almost Linear Function

Let $$f:\mathbb R\to\mathbb R$$ be a function such that $-1\le f(x+y)-f(x)-f(y)\le 1$ for all reals $$x$$, $$y$$. Does there exist a constant $$c$$ such that $$\lvert f(x)-cx\rvert \le 1$$ for all reals $$x$$?

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