In this work, we derive some analytical properties of Berry's phase in one-dimensional quantum and classical crystals, also named Zak's phase. It is commonly assumed in the literature that this phase can only take the values 0 or $\pi$ for a centrosymmetric crystal, however we have found that this assumption is inaccurate and it has its origin in a wrong assumption on Zak's original paper. We provide a general demonstration that Zak's phase can take any value for a non-symmetric crystal but it is strictly zero when it is possible to find a unit cell where the periodic modulation is symmetric. We also demonstrate that Zak's phase is independent of the origin of coordinates selected to compute it. We provide numerical examples verifying this behaviour for both electronic and classical waves (acoustic or photonic). We analyze the weakest electronic potential capable of presenting asymmetry, as well as the double-Dirac delta potential, and in both examples it is found that Zak's phase varies continuously as a function of a symmetry-control parameter, but it is zero when the crystal is symmetric. For classical waves, the layered material is analyzed, and we demonstrate that we need at least three components to have a non-trivial Zak's phase, showing therefore that the binary layered material presents a trivial phase in all the bands of the dispersion diagram. This work shows that Zak's phase and its connection to edge states in one-dimensional crystals should be carefully revisited, since the assumption about its quantization has been widely used in the literature.