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Tracking-by the system output-of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear, single-input, single-output systems modelled by functional differential equations and subject to input saturation. Prespecified is a parameterized performance funnel within which the tracking error is required to evolve; transient and asymptotic behaviour of the tracking error is influenced through choice of parameter values which define the funnel. The control structure is a saturating error feedback with time-varying nonmonotone gain designed to evolve in such a way as to preclude contact with the funnel boundary. A feasibility condition-formulated in bounds of the plant data, the saturation bound, the funnel data, the reference signal, and the initial data-is presented under which the tracking objective is achieved, whilst maintaining boundedness of the state and gain function.