Let $${(G, \cdot)}$$ ( G , · ) be a group and E be a Banach space. Assume that $${f \colon G\rightarrow E}$$ f : G → E is a map such that f(G) is an open set containing 0. If there exists an $${\varepsilon > 0}$$ ε > 0 and a p > 1 so that $$\big| \|f(x) + f(y)\| - \|f(xy)\|\big| \leq \varepsilon \min \big \{\|f(x) + f(y)\|^p, \|f(xy)\|^p\big\}$$ | ‖ f ( x ) + f ( y ) ‖ - ‖ f ( x y ) ‖ | ≤ ε min { ‖ f ( x ) + f ( y ) ‖ p , ‖ f ( x y ) ‖ p } for all $${x, y \in G}$$ x , y ∈ G , then f is an additive map onto E. If E is a finite-dimensional Banach space, the result holds when f(G) (not necessarily open) contains 0 as an interior point.