<p>This paper investigates the stability of linear systems with a time-varying delay, where the delay is assumed to be a bounded function exhibiting non-periodic oscillations with constant amplitude. An improved time-varying Lyapunov functional method is proposed by integrating an interval partitioning technique into the Lyapunov functional framework. In this approach, distinct Lyapunov functionals are constructed over each delay subinterval, and the overall positive definiteness of the time-varying Lyapunov functional is ensured by leveraging the continuity of the functionals at the boundaries between adjacent subintervals. Based on this improved method, new and less conservative stability criteria for time-delay systems are derived. The effectiveness of the proposed approach is demonstrated through two numerical examples.</p>