Lesson plan
Graph continuous function
In mathematics, particularly in game theory and mathematical economics, a function is graph continuous if its graph—the set of all input-output pairs—is a closed set in the product topology of the domain and codomain. In simpler terms, if a sequence of points on the graph converges, its limit point must also belong to the graph. This concept, related to the closed graph property in functional analysis, allows for a broader class of discontinuous payoff functions while enabling equilibrium analysis in economic models. Graph continuity gained prominence through the work of Partha Dasgupta and Eric Maskin in their 1986 paper on the existence of equilibria in discontinuous economic games. Unlike standard continuity, which requires small changes in inputs to produce small changes in outputs, graph continuity permits certain well-behaved discontinuities. This property is crucial for establishing equilibria in settings such as auction theory, oligopoly models, and location competition, where payoff discontinuities naturally arise.
Article summary
In mathematics, particularly in game theory and mathematical economics, a function is graph continuous if its graph—the set of all input-output pairs—is a closed set in the product topology of the domain and codomain. In simpler terms, if a sequence of points on the graph converges, its limit point must also belong to the graph. This concept, related to the closed graph property in functional analysis, allows for a broader class of discontinuous payoff functions while enabling equilibrium analysis in economic models. Graph continuity gained prominence through the work of Partha Dasgupta and Eric Maskin in their 1986 paper on the existence of equilibria in discontinuous economic games. Unlike standard continuity, which requires small changes in inputs to produce small changes in outputs, graph continuity permits certain well-behaved discontinuities. This property is crucial for establishing equilibria in settings such as auction theory, oligopoly models, and location competition, where payoff discontinuities naturally arise.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.