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This paper develops a comprehensive stability analysis framework for continuous-time power control algorithms in wireless networks under bounded time-varying communication delays. Our first set of results establish global asymptotic stability of power control laws involving two-sided scalable interference functions, and include earlier work on standard interference functions as a special case. We then consider contractive interference functions and demonstrate that the associated continuous-time power control laws always have unique fixed points, which are exponentially stable even in the presence of bounded heterogeneous time-varying delays. For this class of interference functions, we derive an explicit bound on the decay rate that allows us to quantify the impact of delays on the convergence time of the algorithm. Numerical simulations illustrate our theoretical results.