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Functional redundancy has often been assumed as an intuitive null hypothesis in biodiversity experiments, but theory based on the classical Lotka-Volterra competition model shows that functional redundancy sensu stricto is incompatible with stable coexistence. Stable coexistence requires differences between species which lead to functional complementarity and differences between the yields of mixtures and monocultures. Only a weaker version of functional redundancy, i.e. that mixture yields lie within the range of variation of monoculture yields, is compatible with stable coexistence in Lotka-Volterra systems. Spatial and temporal environmental variability may provide room for functional redundancy at small spatial and temporal scales, but is not expected to do so at the larger scales at which environmental variations help maintain coexistence. Neutral coexistence of equivalent competitors, non-linear per capita growth rates, and lack of correlation between functional impact and biomass may provide the basis for the existence of functional redundancy in natural ecosystems. Overall, there is a striking parallel between the conditions that allow stable coexistence and those that allow overyielding.