let X be set ; :: thesis: for x0 being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds
f | X is_continuous_in x0
let x0 be Real; :: thesis: for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds
f | X is_continuous_in x0
let S be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds
f | X is_continuous_in x0
let f be PartFunc of REAL, the carrier of S; :: thesis: ( x0 in X & f is_continuous_in x0 implies f | X is_continuous_in x0 )
assume that
A1: x0 in X and
A2: f is_continuous_in x0 ; :: thesis: f | X is_continuous_in x0
A3: x0 in dom f by A2;
A4: dom (f | X) = X /\ (dom f) by RELAT_1:61;
hence A5: x0 in dom (f | X) by A1, A3, XBOOLE_0:def 4; :: according to NFCONT_3:def 1 :: thesis: for s1 being Real_Sequence st rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x0 holds
( (f | X) /* s1 is convergent & (f | X) /. x0 = lim ((f | X) /* s1) )
let s1 be Real_Sequence; :: thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x0 implies ( (f | X) /* s1 is convergent & (f | X) /. x0 = lim ((f | X) /* s1) ) )
assume that
A6: rng s1 c= dom (f | X) and
A7: ( s1 is convergent & lim s1 = x0 ) ; :: thesis: ( (f | X) /* s1 is convergent & (f | X) /. x0 = lim ((f | X) /* s1) )
A8: rng s1 c= dom f by A6, A4, XBOOLE_1:18;
A9: (f | X) /* s1 = f /* s1 by A6, FUNCT_2:117;
hence (f | X) /* s1 is convergent by A2, A7, A8; :: thesis: (f | X) /. x0 = lim ((f | X) /* s1)
x0 in REAL by XREAL_0:def 1;
hence (f | X) /. x0 = f /. x0 by A5, PARTFUN2:15
.= lim ((f | X) /* s1) by A2, A7, A8, A9 ;
:: thesis: verum