:: More on Continuous Functions on Normed Linear Spaces
:: by Hiroyuki Okazaki , Noboru Endou and Yasunari Shidama
::
:: Received August 17, 2010
:: Copyright (c) 2010-2021 Association of Mizar Users

theorem Th1: :: NFCONT_3:1

for n being Nat
for S being RealNormSpace
for seq being Real_Sequence
for h being PartFunc of REAL, the carrier of S st rng seq c= dom h holds
seq . n in dom h

proof end;

theorem Th2: :: NFCONT_3:2

for S being RealNormSpace
for h1, h2 being PartFunc of REAL, the carrier of S
for seq being Real_Sequence st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )

proof end;

theorem :: NFCONT_3:3

for S being RealNormSpace
for h being sequence of S
for r being Real holds r (#) h = r * h

proof end;

theorem Th4: :: NFCONT_3:4

for S being RealNormSpace
for h being PartFunc of REAL, the carrier of S
for seq being Real_Sequence
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)

proof end;

theorem Th5: :: NFCONT_3:5

for S being RealNormSpace
for h being PartFunc of REAL, the carrier of S
for seq being Real_Sequence st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )

proof end;

definition

let S be RealNormSpace;
let f be PartFunc of REAL, the carrier of S;
let x0 be Real;

pred f is_continuous_in x0 means :: NFCONT_3:def 1
( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) );

end;

:: deftheorem defines is_continuous_in NFCONT_3:def 1 :
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S
for x0 being Real holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) );

theorem Th6: :: NFCONT_3:6

for X being set
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

proof end;

theorem :: NFCONT_3:7

for x0 being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Nat holds s1 . n <> x0 ) holds
( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) )

proof end;

theorem Th8: :: NFCONT_3:8

for x0 being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Real st x1 in dom f & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )

proof end;

theorem Th9: :: NFCONT_3:9

for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S
for x0 being Real holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Real st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )

proof end;

theorem Th10: :: NFCONT_3:10

for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S
for x0 being Real holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )

proof end;

theorem :: NFCONT_3:11

for x0 being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0

proof end;

theorem :: NFCONT_3:12

for x0 being Real
for S being RealNormSpace
for f1, f2 being PartFunc of REAL, the carrier of S st x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )

proof end;

theorem Th13: :: NFCONT_3:13

for r, x0 being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st f is_continuous_in x0 holds
r (#) f is_continuous_in x0

proof end;

theorem :: NFCONT_3:14

for x0 being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st x0 in dom f & f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )

proof end;

theorem :: NFCONT_3:15

for x0 being Real
for S, T being RealNormSpace
for f1 being PartFunc of REAL, the carrier of S
for f2 being PartFunc of the carrier of S, the carrier of T st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds
f2 * f1 is_continuous_in x0

proof end;

definition

let S be RealNormSpace;
let f be PartFunc of REAL, the carrier of S;

attr f is continuous means :Def2: :: NFCONT_3:def 2
for x0 being Real st x0 in dom f holds
f is_continuous_in x0;

end;

:: deftheorem Def2 defines continuous NFCONT_3:def 2 :
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S holds
( f is continuous iff for x0 being Real st x0 in dom f holds
f is_continuous_in x0 );

theorem Th16: :: NFCONT_3:16

for S being RealNormSpace
for X being set
for f being PartFunc of REAL, the carrier of S st X c= dom f holds
( f | X is continuous iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) )

proof end;

theorem Th17: :: NFCONT_3:17

for X being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st X c= dom f holds
( f | X is continuous iff for x0, r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Real st x1 in X & |.(x1 - x0).| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )

proof end;

registration

let S be RealNormSpace;

cluster Function-like constant -> continuous for Element of bool [:REAL, the carrier of S:];

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ is constant holds
b₁ is continuous

proof end;

end;

registration

let S be RealNormSpace;

cluster Relation-like REAL -defined the carrier of S -valued Function-like continuous for Element of bool [:REAL, the carrier of S:];

existence
ex b₁ being PartFunc of REAL, the carrier of S st b₁ is continuous

proof end;

end;

registration

let S be RealNormSpace;
let f be continuous PartFunc of REAL, the carrier of S;
let X be set ;

cluster f | X -> continuous for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f | X holds
b₁ is continuous

proof end;

end;

theorem Th18: :: NFCONT_3:18

for X, X1 being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st f | X is continuous & X1 c= X holds
f | X1 is continuous

proof end;

registration

let S be RealNormSpace;

cluster empty Function-like -> continuous for Element of bool [:REAL, the carrier of S:];

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ is empty holds
b₁ is continuous ;

end;

registration

let S be RealNormSpace;
let f be PartFunc of REAL, the carrier of S;
let X be trivial set ;

cluster f | X -> continuous for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f | X holds
b₁ is continuous

proof end;

end;

registration

let S be RealNormSpace;
let f1, f2 be continuous PartFunc of REAL, the carrier of S;

cluster f1 + f2 -> continuous for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f1 + f2 holds
b₁ is continuous

proof end;

cluster f1 - f2 -> continuous for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f1 - f2 holds
b₁ is continuous

proof end;

end;

theorem Th19: :: NFCONT_3:19

for S being RealNormSpace
for X being set
for f1, f2 being PartFunc of REAL, the carrier of S st X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous holds
( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous )

proof end;

theorem :: NFCONT_3:20

for S being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of REAL, the carrier of S st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds
( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous )

proof end;

registration

let S be RealNormSpace;
let f be continuous PartFunc of REAL, the carrier of S;
let r be Real;

cluster r (#) f -> continuous for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = r (#) f holds
b₁ is continuous

proof end;

end;

theorem Th21: :: NFCONT_3:21

for X being set
for r being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds
(r (#) f) | X is continuous

proof end;

theorem :: NFCONT_3:22

for X being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds
( ||.f.|| | X is continuous & (- f) | X is continuous )

proof end;

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

proof end;

theorem Th24: :: NFCONT_3:24

for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st dom f is compact & f | (dom f) is continuous holds
rng f is compact

proof end;

theorem :: NFCONT_3:25

for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S
for Y being Subset of REAL st Y c= dom f & Y is compact & f | Y is continuous holds
f .: Y is compact

proof end;

definition

let S be RealNormSpace;
let f be PartFunc of REAL, the carrier of S;

attr f is Lipschitzian means :Def3: :: NFCONT_3:def 3
ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom f & x2 in dom f holds
||.((f /. x1) - (f /. x2)).|| <= r * |.(x1 - x2).| ) );

end;

:: deftheorem Def3 defines Lipschitzian NFCONT_3:def 3 :
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S holds
( f is Lipschitzian iff ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom f & x2 in dom f holds
||.((f /. x1) - (f /. x2)).|| <= r * |.(x1 - x2).| ) ) );

theorem Th26: :: NFCONT_3:26

for X being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S holds
( f | X is Lipschitzian iff ex r being Real st
( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds
||.((f /. x1) - (f /. x2)).|| <= r * |.(x1 - x2).| ) ) )

proof end;

registration

let S be RealNormSpace;

cluster empty Function-like -> Lipschitzian for Element of bool [:REAL, the carrier of S:];

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ is empty holds
b₁ is Lipschitzian

proof end;

end;

registration

let S be RealNormSpace;

cluster Relation-like REAL -defined the carrier of S -valued empty Function-like for Element of bool [:REAL, the carrier of S:];

existence
ex b₁ being PartFunc of REAL, the carrier of S st b₁ is empty

proof end;

end;

registration

let S be RealNormSpace;
let f be Lipschitzian PartFunc of REAL, the carrier of S;
let X be set ;

cluster f | X -> Lipschitzian for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f | X holds
b₁ is Lipschitzian

proof end;

end;

theorem :: NFCONT_3:27

for X, X1 being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X1 c= X holds
f | X1 is Lipschitzian

proof end;

registration

let S be RealNormSpace;
let f1, f2 be Lipschitzian PartFunc of REAL, the carrier of S;

cluster f1 + f2 -> Lipschitzian for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f1 + f2 holds
b₁ is Lipschitzian

proof end;

cluster f1 - f2 -> Lipschitzian for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = f1 - f2 holds
b₁ is Lipschitzian

proof end;

end;

theorem :: NFCONT_3:28

for X, X1 being set
for S being RealNormSpace
for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds
(f1 + f2) | (X /\ X1) is Lipschitzian

proof end;

theorem :: NFCONT_3:29

for X, X1 being set
for S being RealNormSpace
for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds
(f1 - f2) | (X /\ X1) is Lipschitzian

proof end;

registration

let S be RealNormSpace;
let f be Lipschitzian PartFunc of REAL, the carrier of S;
let p be Real;

cluster p (#) f -> Lipschitzian for PartFunc of REAL, the carrier of S;

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ = p (#) f holds
b₁ is Lipschitzian

proof end;

end;

theorem :: NFCONT_3:30

for X being set
for p being Real
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X c= dom f holds
(p (#) f) | X is Lipschitzian

proof end;

registration

let S be RealNormSpace;
let f be Lipschitzian PartFunc of REAL, the carrier of S;

cluster ||.f.|| -> Lipschitzian for PartFunc of REAL,REAL;

coherence
for b₁ being PartFunc of REAL,REAL st b₁ = ||.f.|| holds
b₁ is Lipschitzian

proof end;

end;

theorem :: NFCONT_3:31

for X being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian holds
( - (f | X) is Lipschitzian & (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian )

proof end;

registration

let S be RealNormSpace;

cluster Function-like constant -> Lipschitzian for Element of bool [:REAL, the carrier of S:];

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ is constant holds
b₁ is Lipschitzian

proof end;

end;

registration

let S be RealNormSpace;

cluster Function-like Lipschitzian -> continuous for Element of bool [:REAL, the carrier of S:];

coherence
for b₁ being PartFunc of REAL, the carrier of S st b₁ is Lipschitzian holds
b₁ is continuous

proof end;

end;

theorem :: NFCONT_3:32

for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S st ex r being Point of S st rng f = {r} holds
f is continuous

proof end;

theorem :: NFCONT_3:33

for X being set
for S being RealNormSpace
for f being PartFunc of REAL, the carrier of S
for r, p being Point of S st ( for x0 being Real st x0 in X holds
f /. x0 = (x0 * r) + p ) holds
f | X is continuous

proof end;