theorem :: FCONT_2:20

for g, p being Real
for f being one-to-one PartFunc of REAL,REAL st p <= g & [.p,g.] c= dom f & f | [.p,g.] is continuous holds
(f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous