:: Real Function Continuity
:: by Konrad Raczkowski and Pawe{\l} Sadowski
::
:: Received June 18, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let f be PartFunc of REAL,REAL;
let x0 be Real;

pred f is_continuous_in x0 means :: FCONT_1:def 1
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 FCONT_1:def 1 :
for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_continuous_in x0 iff 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 Th1: :: FCONT_1:1

for X being set
for x0 being Real
for f being PartFunc of REAL,REAL st x0 in X & f is_continuous_in x0 holds
f | X is_continuous_in x0

proof end;

theorem :: FCONT_1:2

for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_continuous_in x0 iff 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 Th3: :: FCONT_1:3

for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_continuous_in x0 iff 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 Th4: :: FCONT_1:4

for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_continuous_in x0 iff 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 Th5: :: FCONT_1:5

for f being PartFunc of REAL,REAL
for x0 being Real holds
( f is_continuous_in x0 iff for N1 being Neighbourhood of f . x0 ex N being Neighbourhood of x0 st f .: N c= N1 )

proof end;

theorem :: FCONT_1:6

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

proof end;

theorem Th7: :: FCONT_1:7

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL 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 & f1 (#) f2 is_continuous_in x0 )

proof end;

theorem Th8: :: FCONT_1:8

for r, x0 being Real
for f being PartFunc of REAL,REAL st x0 in dom f & f is_continuous_in x0 holds
r (#) f is_continuous_in x0

proof end;

theorem :: FCONT_1:9

for x0 being Real
for f being PartFunc of REAL,REAL st x0 in dom f & f is_continuous_in x0 holds
( abs f is_continuous_in x0 & - f is_continuous_in x0 )

proof end;

theorem Th10: :: FCONT_1:10

for x0 being Real
for f being PartFunc of REAL,REAL st f is_continuous_in x0 & f . x0 <> 0 holds
f ^ is_continuous_in x0

proof end;

theorem :: FCONT_1:11

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st x0 in dom f2 & f1 is_continuous_in x0 & f1 . x0 <> 0 & f2 is_continuous_in x0 holds
f2 / f1 is_continuous_in x0

proof end;

theorem Th12: :: FCONT_1:12

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL 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 f be PartFunc of REAL,REAL;

attr f is continuous means :Def2: :: FCONT_1:def 2
for x0 being Real st x0 in dom f holds
f is_continuous_in x0;

end;

:: deftheorem Def2 defines continuous FCONT_1:def 2 :
for f being PartFunc of REAL,REAL holds
( f is continuous iff for x0 being Real st x0 in dom f holds
f is_continuous_in x0 );

theorem Th13: :: FCONT_1:13

for X being set
for f being PartFunc of REAL,REAL 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 Th14: :: FCONT_1:14

for X being set
for f being PartFunc of REAL,REAL 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

cluster Function-like V8() -> continuous for Element of bool [:REAL,REAL:];

coherence
for b₁ being PartFunc of REAL,REAL st b₁ is V8() holds
b₁ is continuous

proof end;

end;

registration

cluster Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued continuous for Element of bool [:REAL,REAL:];

existence
ex b₁ being PartFunc of REAL,REAL st b₁ is continuous

proof end;

end;

registration

let f be continuous PartFunc of REAL,REAL;
let X be set ;

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

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

proof end;

end;

theorem :: FCONT_1:15

for X being set
for f being PartFunc of REAL,REAL holds
( f | X is continuous iff (f | X) | X is continuous ) ;

theorem Th16: :: FCONT_1:16

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

proof end;

registration

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

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

end;

registration

let f be PartFunc of REAL,REAL;
let X be trivial set ;

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

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

proof end;

end;

theorem :: FCONT_1:17

for x0 being Real
for f being PartFunc of REAL,REAL holds f | {x0} is continuous ;

registration

let f1, f2 be continuous PartFunc of REAL,REAL;

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

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

proof end;

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

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

proof end;

cluster f1 (#) f2 -> continuous for PartFunc of REAL,REAL;

coherence
for b₁ being PartFunc of REAL,REAL st b₁ = f1 (#) f2 holds
b₁ is continuous

proof end;

end;

theorem Th18: :: FCONT_1:18

for X being set
for f1, f2 being PartFunc of REAL,REAL 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 & (f1 (#) f2) | X is continuous )

proof end;

theorem :: FCONT_1:19

for X, X1 being set
for f1, f2 being PartFunc of REAL,REAL 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 & (f1 (#) f2) | (X /\ X1) is continuous )

proof end;

registration

let f be continuous PartFunc of REAL,REAL;
let r be Real;

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

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

proof end;

end;

theorem Th20: :: FCONT_1:20

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

proof end;

theorem :: FCONT_1:21

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is continuous holds
( (abs f) | X is continuous & (- f) | X is continuous )

proof end;

theorem Th22: :: FCONT_1:22

for X being set
for f being PartFunc of REAL,REAL st f | X is continuous & f " {0} = {} holds
(f ^) | X is continuous

proof end;

theorem :: FCONT_1:23

for X being set
for f being PartFunc of REAL,REAL st f | X is continuous & (f | X) " {0} = {} holds
(f ^) | X is continuous

proof end;

theorem :: FCONT_1:24

for X being set
for f1, f2 being PartFunc of REAL,REAL st X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f1 " {0} = {} & f2 | X is continuous holds
(f2 / f1) | X is continuous

proof end;

registration

let f1, f2 be continuous PartFunc of REAL,REAL;

cluster f1 (#) f2 -> continuous for PartFunc of REAL,REAL;

coherence
for b₁ being PartFunc of REAL,REAL st b₁ = f2 * f1 holds
b₁ is continuous

proof end;

end;

theorem :: FCONT_1:25

for X being set
for f1, f2 being PartFunc of REAL,REAL st f1 | X is continuous & f2 | (f1 .: X) is continuous holds
(f2 * f1) | X is continuous

proof end;

theorem :: FCONT_1:26

for X, X1 being set
for f1, f2 being PartFunc of REAL,REAL st f1 | X is continuous & f2 | X1 is continuous holds
(f2 * f1) | (X /\ (f1 " X1)) is continuous

proof end;

theorem :: FCONT_1:27

for f being PartFunc of REAL,REAL 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 Th28: :: FCONT_1:28

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

proof end;

theorem :: FCONT_1:29

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

proof end;

theorem Th30: :: FCONT_1:30

for f being PartFunc of REAL,REAL st dom f <> {} & dom f is compact & f | (dom f) is continuous holds
ex x1, x2 being Real st
( x1 in dom f & x2 in dom f & f . x1 = upper_bound (rng f) & f . x2 = lower_bound (rng f) )

proof end;

:: Extreme value theorem

theorem :: FCONT_1:31

for f being PartFunc of REAL,REAL
for Y being Subset of REAL st Y <> {} & Y c= dom f & Y is compact & f | Y is continuous holds
ex x1, x2 being Real st
( x1 in Y & x2 in Y & f . x1 = upper_bound (f .: Y) & f . x2 = lower_bound (f .: Y) )

proof end;

definition

let f be PartFunc of REAL,REAL;

attr f is Lipschitzian means :Def3: :: FCONT_1: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 FCONT_1:def 3 :
for f being PartFunc of REAL,REAL 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 Th32: :: FCONT_1:32

for X being set
for f being PartFunc of REAL,REAL 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

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

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

proof end;

end;

registration

cluster Relation-like REAL -defined REAL -valued Function-like empty complex-valued ext-real-valued real-valued for Element of bool [:REAL,REAL:];

existence
ex b₁ being PartFunc of REAL,REAL st b₁ is empty

proof end;

end;

registration

let f be Lipschitzian PartFunc of REAL,REAL;
let X be set ;

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

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

proof end;

end;

theorem :: FCONT_1:33

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

proof end;

registration

let f1, f2 be Lipschitzian PartFunc of REAL,REAL;

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

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

proof end;

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

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

proof end;

end;

theorem :: FCONT_1:34

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

proof end;

theorem :: FCONT_1:35

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

proof end;

registration

let f1, f2 be bounded Lipschitzian PartFunc of REAL,REAL;

cluster f1 (#) f2 -> Lipschitzian for PartFunc of REAL,REAL;

coherence
for b₁ being PartFunc of REAL,REAL st b₁ = f1 (#) f2 holds
b₁ is Lipschitzian

proof end;

end;

theorem :: FCONT_1:36

for X, X1, Z, Z1 being set
for f1, f2 being PartFunc of REAL,REAL st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian & f1 | Z is bounded & f2 | Z1 is bounded holds
(f1 (#) f2) | (((X /\ Z) /\ X1) /\ Z1) is Lipschitzian

proof end;

registration

let f be Lipschitzian PartFunc of REAL,REAL;
let p be Real;

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

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

proof end;

end;

theorem :: FCONT_1:37

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

proof end;

registration

let f be Lipschitzian PartFunc of REAL,REAL;

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

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

proof end;

end;

theorem :: FCONT_1:38

for X being set
for f being PartFunc of REAL,REAL st f | X is Lipschitzian holds
( - (f | X) is Lipschitzian & (abs f) | X is Lipschitzian )

proof end;

registration

cluster Function-like V8() -> Lipschitzian for Element of bool [:REAL,REAL:];

coherence
for b₁ being PartFunc of REAL,REAL st b₁ is V8() holds
b₁ is Lipschitzian

proof end;

end;

registration

let Y be Subset of REAL;

cluster id Y -> Lipschitzian for PartFunc of REAL,REAL;

coherence
for b₁ being PartFunc of REAL,REAL st b₁ = id Y holds
b₁ is Lipschitzian

proof end;

end;

registration

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

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

proof end;

end;

theorem :: FCONT_1:39

for f being PartFunc of REAL,REAL st ex r being Real st rng f = {r} holds
f is continuous

proof end;

theorem :: FCONT_1:40

for f being PartFunc of REAL,REAL st ( for x0 being Real st x0 in dom f holds
f . x0 = x0 ) holds
f is continuous

proof end;

theorem Th41: :: FCONT_1:41

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

proof end;

theorem Th42: :: FCONT_1:42

for f being PartFunc of REAL,REAL st ( for x0 being Real st x0 in dom f holds
f . x0 = x0 ^2 ) holds
f | (dom f) is continuous

proof end;

theorem :: FCONT_1:43

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & ( for x0 being Real st x0 in X holds
f . x0 = x0 ^2 ) holds
f | X is continuous

proof end;

theorem Th44: :: FCONT_1:44

for f being PartFunc of REAL,REAL st ( for x0 being Real st x0 in dom f holds
f . x0 = |.x0.| ) holds
f is continuous

proof end;

theorem :: FCONT_1:45

for X being set
for f being PartFunc of REAL,REAL st ( for x0 being Real st x0 in X holds
f . x0 = |.x0.| ) holds
f | X is continuous

proof end;

theorem Th46: :: FCONT_1:46

for X being set
for f being PartFunc of REAL,REAL st f | X is monotone & ex p, g being Real st
( p <= g & f .: X = [.p,g.] ) holds
f | X is continuous

proof end;

theorem :: FCONT_1:47

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 increasing or f | [.p,g.] is decreasing ) holds
((f | [.p,g.]) ") | (f .: [.p,g.]) is continuous

proof end;

:: from

definition

let a, b be Real;

func AffineMap (a,b) -> Function of REAL,REAL means :Def4: :: FCONT_1:def 4
for x being Real holds it . x = (a * x) + b;

existence
ex b₁ being Function of REAL,REAL st
for x being Real holds b₁ . x = (a * x) + b

proof end;

uniqueness
for b₁, b₂ being Function of REAL,REAL st ( for x being Real holds b₁ . x = (a * x) + b ) & ( for x being Real holds b₂ . x = (a * x) + b ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines AffineMap FCONT_1:def 4 :
for a, b being Real
for b₃ being Function of REAL,REAL holds
( b₃ = AffineMap (a,b) iff for x being Real holds b₃ . x = (a * x) + b );

registration

let a, b be Real;

cluster AffineMap (a,b) -> continuous ;

coherence
AffineMap (a,b) is continuous

proof end;

end;

registration

cluster Relation-like REAL -defined REAL -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued continuous for Element of bool [:REAL,REAL:];

existence
ex b₁ being Function of REAL,REAL st b₁ is continuous

proof end;

end;

theorem Th48: :: FCONT_1:48

for a, b being Real holds (AffineMap (a,b)) . 0 = b

proof end;

theorem Th49: :: FCONT_1:49

for a, b being Real holds (AffineMap (a,b)) . 1 = a + b

proof end;

theorem Th50: :: FCONT_1:50

for a, b being Real st a <> 0 holds
AffineMap (a,b) is one-to-one

proof end;

theorem :: FCONT_1:51

for a, b, x, y being Real st a > 0 & x < y holds
(AffineMap (a,b)) . x < (AffineMap (a,b)) . y

proof end;

theorem :: FCONT_1:52

for a, b, x, y being Real st a < 0 & x < y holds
(AffineMap (a,b)) . x > (AffineMap (a,b)) . y

proof end;

theorem Th53: :: FCONT_1:53

for a, b, x, y being Real st a >= 0 & x <= y holds
(AffineMap (a,b)) . x <= (AffineMap (a,b)) . y

proof end;

theorem :: FCONT_1:54

for a, b, x, y being Real st a <= 0 & x <= y holds
(AffineMap (a,b)) . x >= (AffineMap (a,b)) . y

proof end;

theorem Th55: :: FCONT_1:55

for a, b being Real st a <> 0 holds
rng (AffineMap (a,b)) = REAL

proof end;

theorem :: FCONT_1:56

for a, b being Real st a <> 0 holds
(AffineMap (a,b)) " = AffineMap ((a "),(- (b / a)))

proof end;

theorem :: FCONT_1:57

for a, b being Real st a > 0 holds
(AffineMap (a,b)) .: [.0,1.] = [.b,(a + b).]

proof end;