theorem :: NFCONT_3:23

for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st f is total & ( for x1, x2 being Real holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Real st f is_continuous_in x0 holds
f | REAL is continuous