linear algebra - What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$,
$$f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$}$$
Q1. I'd like to know how weak one can make additional assumptions to guarantee that $f$ is affine, namely that there exists a linear transformation $L:V\to V$, and an element $v\in V$ such that
$$\begin{align} f(x) = L(x) + v \end{align}$$
for all $x\in V$.

I think, for example, that if one assumes that $V = \mathbb R^n$, and if $f$ is differentiable, then $(\star)$ implies that $f$ is affine. This makes me think that perhaps one needs a notion of smoothness or continuity in general and therefore possibly a norm? Also

Q2. Is there a common name for the property $(\star)$?