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*f*is continuous if*f*^{ -1}(*G*) is open whenever*G*isThat 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

continued...

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