This work investigates an infinite-horizon optimal consumption and portfolio selection problem in a continuous-time market, where an agent’s utility is derived from the difference between their current consumption rate and a path-dependent reference level. The reference level is modeled as a scaled running maximum of historical consumption, capturing the psychological impact of habit persistence. While recent literature has primarily explored this preference structure under specific utility forms such as exponential utility, we generalize the analysis to a broad class of utility functions satisfying standard asymptotic elasticity conditions. To handle the inherent path-dependency and non-negativity constraints on consumption, we develop a comprehensive convex martingale duality framework.We demonstrate that the optimal consumption policy naturally partitions the dual state space into four distinct structural regimes, including an active singular reflection phase where the reference maximum is dynamically updated. We characterize the dual value function as a viscosity solution to a free-boundary Variational Inequality (VI). By establishing its strict equivalence to an auxiliary optimal stopping problem, we explicitly construct a unique, piecewise C^2 closed-form solution, which exactly coincides with the dual value function.