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Real Analysis : Topological Characterization of Continuity

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Let g be defined on all of R.if A is a subset of R define the set g^-1 (A) by g^-1 (A)={x belong to R:g(x) belong to A}. Show that g is continous if and only if g^-1 (O) is open whenever O subset or equal to R is an open set.

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Solution Summary

The topological characterization of continuity is investigated. The solution is concise.

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Proof:

"=>" If g is continuous, we consider an arbitrary x in g^-1(O), where O is an open subset of R. Then we know g(x) belongs to O. Since O is an open set, we can find some e>0, such that the interval(g(x)-e,g(x)+e) belongs to O. For this e, ...

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