On the Scottish Book Problem 155 by Mazur and Sternbach

Problem 155 of the Scottish Book asks whether every bijection $U\text{\hspace{0.17em}⁣}:X\to Y$ between two Banach spaces $X,Y$ with the property that, each point of $X$ has a neighborhood on which $U$ is isometric, is globally isometric on $X$. We prove that this is true under the additional assumption that $X$ is separable and the weaker assumption of surjectivity instead of bijectivity.