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*f*is continuous if, for open*G*,*f*^{ -1}(*G*) is openThat is, if

*f*^{ -1}(*G*) is semidecidable whenever*G*is semidecidableSay

*f*is discontinuousThen there is a semidecidable

*G*for which*f*^{ -1}(*G*) is not semidecidableBut here's an algorithm for semideciding

*f*^{ -1}(*G*):

if G(f(x)): then YES else NO

The only way this can fail is if f(x)

*is not computable*

Open set | = | Semidecidable property |

Closed set | = | Semidecidable complement |

Clopen set | = | Decidable property |

Continuous function | = | Computable function |

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