let S be RealNormSpace; :: thesis: for x0 being Point of S
for f, g being PartFunc of S,RNS_Real st f is_continuous_in x0 & g = ||.f.|| holds
g is_continuous_in x0
let x0 be Point of S; :: thesis: for f, g being PartFunc of S,RNS_Real st f is_continuous_in x0 & g = ||.f.|| holds
g is_continuous_in x0
let f, g be PartFunc of S,RNS_Real; :: thesis: ( f is_continuous_in x0 & g = ||.f.|| implies g is_continuous_in x0 )
assume that
A1: f is_continuous_in x0 and
A2: g = ||.f.|| ; :: thesis: g is_continuous_in x0
A3: ||.f.|| is_continuous_in x0 by A1, NFCONT_1:17;
for s1 being sequence of S st rng s1 c= dom g & s1 is convergent & lim s1 = x0 holds
( g /* s1 is convergent & g /. x0 = lim (g /* s1) )

proof

let s1 be sequence of S; :: thesis: ( rng s1 c= dom g & s1 is convergent & lim s1 = x0 implies ( g /* s1 is convergent & g /. x0 = lim (g /* s1) ) )
assume that
A4: rng s1 c= dom g and
A5: s1 is convergent and
A6: lim s1 = x0 ; :: thesis: ( g /* s1 is convergent & g /. x0 = lim (g /* s1) )
( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) ) by A3, A2, A4, A5, A6;
hence ( g /* s1 is convergent & g /. x0 = lim (g /* s1) ) by A2, DUALSP03:5, DUALSP03:6; :: thesis: verum

end;

hence g is_continuous_in x0 by A2, A1, NORMSP_0:def 2; :: thesis: verum