Continuity

• f is continuous if, for open G, f ^-1(G) is open

• That is, if f ^-1(G) is semidecidable whenever G is semidecidable

• Say f is discontinuous

• Then there is a semidecidable G for which f ^-1(G) is not semidecidable

• But 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