This paper investigates the limit state of an elastic strip composed of a heterogeneous material with uneven side surfaces. Compressive forces are considered independently along the upper and lower boundaries as well as the lateral edges of the strip’s cross-section. A criterion based on the continuous dependence of the system’s response on initial data is proposed as a necessary condition for identifying the disruption of normal functioning. A violation of this continuity can lead to two types of limit states: the first involving a loss of stability, and the second characterized by excessive deformations and potential system failure. In the mathematical model, boundary conditions in the deformed configuration are incorporated, and the influence of rotation angles in the equilibrium equations is taken into account following the approaches of Novozhilov and Ishlinsky. A condition is derived that identifies the boundary region where the strip reaches a limit state, corresponding to the loss of stability of its equilibrium form. The impact of nonlinearity in the equilibrium equations within this critical region is also analyzed. The reliability of the results is supported by their agreement with established findings in the literature. Additionally, for various cross-sectional parameter values, regions are constructed where the stress–strain state remains approximately uniform.