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This paper provides a theoretical framework for the analysis of interference-coupled multiuser systems. The fundamental behavior of such a system is modeled by interference functions, defined by axioms ldquononnegativity, rdquoscale-invariance,rdquo and ldquomonotonicity.rdquo It is shown that every interference function has an interpretation as the optimum of a min-max problem, where the optimization is over a closed comprehensive positive coefficient set. This provides new insight into the structure of general interference functions and its elementary building blocks. Conversely, it is shown that every closed comprehensive positive set can be expressed as a level set of an interference function. This shows a close connection between the analysis of interference functions and multiuser performance regions, which are typically closed comprehensive. The generality of this framework allows for a wide range of potential applications. As an example, we analyze the problem of interference balancing.