theorem Th10: :: NCFCONT1:10

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