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The aim of this paper is to propose a new way to deal with observability of systems goved by ODEs,in a more general setting than the standard output equation.The primary finding is that observability over a time horizon reduces to single-valuedness of the vertical section of a set we name the observability kel.The latter consists of the viability kel of the output domain under the augmented system.The approach may be used either for global or local observability,to which available results on single-valuedness of multifunctions shall be applied in order to get necessary and/or sufficient characterizing conditions.Several examples are provided in order to illustrate the method.