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This paper proposes a basis-function canonical piecewise-linear (BF-CPWL) function, which can approximate any continuous function using a weighted sum of PWL BFs. The BF-CPWL approximation integrates Breiman's hinging hyperplane model and Julian's high-level canonical PWL approximation into a common theoretical framework. Moreover, an approximation algorithm is developed, which fits and adds the PWL BFs iteratively using a modified Gauss-Newton method. This algorithm guarantees a local convergence, while achieving a good tradeoff between computational simplicity and approximation accuracy. The BF-CPWL approximation can find applications in nonlinear circuit synthesis, dynamic system identification and control.