theorem Th45: :: NCFCONT1:45

for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )