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The system of general hybrid orthogonal functions (GHOF) is shown to be more relevant to many practical situations than the conventional systems of orthogonal functions. The proposed framework of definition combines the natural features of discontinuity of the class of piecewise-constant systems and the inherent characteristics of continuity and differentiability of the systems of continuous systems of polynomial and sinusoidal functions, leading to a general, flexible and piecewise-continuous (and differentiable) system of GHOF. The powerfulness of the proposed system is demonstrated in the representation of functions with discontinuities, solution of the state equation with discontinuous inputs and prediction of the limit cycle of a highly nonlinear van der Pol's oscillator. The results amply demonstrate the value of the GHOF as a potential basis for function expansion in a large class of real-world problems.