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This brief proposes a general piecewise-linear (GPWL) representation theorem based on geometrical structure analysis of continuous PWL functions. A constructive algorithm is developed to represent a continuous PWL function with the weighted sum of GPWL basis functions. The GPWL basis functions are defined over a domain partition with pairwise directly adjacent regions. The GPWL representation theorem unifies many known PWL models into a common theoretical framework. The GPWL representation is promising to find applications in nonlinear circuit synthesis, dynamic system identification, and control.