theorem :: NCFCONT1:76

for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st f is total & ( for x1, x2 being Point of RNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of RNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of RNS