Question: Is the coexistence of a continuum of species or ecological types possible in real-world communities? Or should one expect distinctly different species? 

Mathematical methods: We study whether the coexistence of species in a continuum of ecological types is (a) dynamically stable (against changes in population densities) and (b) structurally robust (against changes in population dynamics). Since most of the reviewed investigations are based on Lotka-Volterra models, we carefully explain which of the presented conclusions are model-independent. 

mathematical conclusions: Seemingly plausible models with dynamically stable continuous- coexistence solutions do exist. However, these models either depend on biologically unrealistic mathematical assumptions (e.g. non-differentiable ingredient functions) or are structurally unstable (i.e. destroyable by arbitrarily small modifications to those ingredient functions). The dynamical stability of a continuous-coexistence solution, if it exists, requires positive definiteness of the model's competition kernel. 

Biological conclusions:  While the classical expectation of fixed limits to similarity is mathematically naive, the fundamental discreteness of species is a natural consequence of the basic structure of ecological interaction