Can one recognize a function from its graph? (English) Zbl 1529.54004
In the present paper the authors want to answer the following two dual questions.
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- Given a class of functions \(f\colon X\to Y\), where \(X\) and \(Y\) are metric spaces, can we find a property of their graphs as subsets of \(X\times Y\) which exactly characterizes functions from this class?
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- Given a property of an appropriate subset of \(X\times Y\), can we describe the exact class of functions \(f\colon X\to Y\) such that their graphs have this property?
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- functions with closed graphs;
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- functions with connected graphs;
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- functions with pathwise connected graphs;
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- regular Darboux functions.
If the graphs of two functions \(f\colon X\to Y\) and \(g\colon Y\to Z\) have a certain topological property, does the graph of their composition \(g\circ f\colon X\to Z\) have the same property?
They give some answers on this problem concerning the classes of functions with closed graphs, functions with connected graphs and functions which have closed fibres. Throughout the paper there are many examples and counterexamples illustrating the relationships between the discussed classes of functions.
Reviewer: Waldemar Sieg (Bydgoszcz)
MSC:
| 54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |
| 26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
| 54C50 | Topology of special sets defined by functions |
Keywords:
continuous functions; Darboux continuous functions; Baire class one functions; closed graph; connected graph; pathwise connected graph; locally connected graph; \(F_{\sigma}\) graph; \(G_{\delta}\) graphReferences:
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