 Solution

The vector field is continuous on the cube. Part (b) gives $M=1$, and part (c) supplies a uniform Lipschitz constant $L=1$. In the quantitative Picard-Lindelöf theorem, the independent-variable half-width and the dependent-variable radius are both one, soThereforeThe global derivative bound also makes the cube condition explicit. Along any solution segment starting at the initial point,Thus it cannot leave the dependent-variable ball before $|x|=1$. Bounded derivatives give limiting endpoint values, and local existence extends to them because $F$ is defined continuously on all space. Uniform Lipschitz continuity ensures that all such local pieces agree on overlaps, proving uniqueness throughout the closed interval.