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Zero-point energy/ground state energy is simply the lowest possible energy that a quantum mechanical system may have. Unlike what we study in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. Coming to continuity of wave function, a particle is defined by a wave function , Be-2x for x<0 and Ce4x for x>0. For the wave function to be continuous at x=0, B=C. A wave function must be continuous for it to be valid. So ,  wave function has to be continuous at all points, and there's  no exception.

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