:: Monotonic and Continuous Real Function
:: by Jaros{\l}aw Kotowicz
::
:: Received January 10, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

registration

cluster [#] REAL -> closed ;

coherence
[#] REAL is closed by XREAL_0:def 1;

cluster empty -> closed for Element of K19(REAL);

coherence
for b₁ being Subset of REAL st b₁ is empty holds
b₁ is closed by XBOOLE_1:3;

end;

registration

cluster [#] REAL -> open ;

coherence
[#] REAL is open

proof end;

cluster empty -> open for Element of K19(REAL);

coherence
for b₁ being Subset of REAL st b₁ is empty holds
b₁ is open

proof end;

end;

registration

let r be Real;

cluster right_closed_halfline r -> closed ;

coherence
right_closed_halfline r is closed

proof end;

cluster left_closed_halfline r -> closed ;

coherence
left_closed_halfline r is closed

proof end;

cluster right_open_halfline r -> open ;

coherence
right_open_halfline r is open

proof end;

cluster halfline r -> open ;

coherence
halfline r is open

proof end;

end;

theorem Th1: :: FCONT_3:1

for x0, g, r being Real st g in ].(x0 - r),(x0 + r).[ holds
|.(g - x0).| < r

proof end;

theorem :: FCONT_3:2

for x0, g, r being Real st g in ].(x0 - r),(x0 + r).[ holds
g - x0 in ].(- r),r.[

proof end;

theorem Th3: :: FCONT_3:3

for x0, r1, g, r being Real st g = x0 + r1 & |.r1.| < r holds
( 0 < r & g in ].(x0 - r),(x0 + r).[ )

proof end;

theorem :: FCONT_3:4

for x0, g, r being Real st g - x0 in ].(- r),r.[ holds
( 0 < r & g in ].(x0 - r),(x0 + r).[ )

proof end;

theorem Th5: :: FCONT_3:5

for p being Real
for a being Real_Sequence
for x0 being Real st ( for n being Nat holds a . n = x0 - (p / (n + 1)) ) holds
( a is convergent & lim a = x0 )

proof end;

theorem Th6: :: FCONT_3:6

for p being Real
for a being Real_Sequence
for x0 being Real st ( for n being Nat holds a . n = x0 + (p / (n + 1)) ) holds
( a is convergent & lim a = x0 )

proof end;

theorem :: FCONT_3:7

for x0, r being Real
for f being PartFunc of REAL,REAL st f is_continuous_in x0 & f . x0 <> r & ex N being Neighbourhood of x0 st N c= dom f holds
ex N being Neighbourhood of x0 st
( N c= dom f & ( for g being Real st g in N holds
f . g <> r ) )

proof end;

theorem :: FCONT_3:8

for X being set
for f being PartFunc of REAL,REAL st ( f | X is increasing or f | X is decreasing ) holds
f | X is one-to-one

proof end;

theorem Th9: :: FCONT_3:9

for X being set
for f being one-to-one PartFunc of REAL,REAL st f | X is increasing holds
((f | X) ") | (f .: X) is increasing

proof end;

theorem Th10: :: FCONT_3:10

for X being set
for f being one-to-one PartFunc of REAL,REAL st f | X is decreasing holds
((f | X) ") | (f .: X) is decreasing

proof end;

theorem Th11: :: FCONT_3:11

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is monotone & ex p being Real st f .: X = left_open_halfline p holds
f | X is continuous

proof end;

theorem Th12: :: FCONT_3:12

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is monotone & ex p being Real st f .: X = right_open_halfline p holds
f | X is continuous

proof end;

theorem Th13: :: FCONT_3:13

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is monotone & ex p being Real st f .: X = left_closed_halfline p holds
f | X is continuous

proof end;

theorem Th14: :: FCONT_3:14

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is monotone & ex p being Real st f .: X = right_closed_halfline p holds
f | X is continuous

proof end;

theorem Th15: :: FCONT_3:15

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

proof end;

theorem Th16: :: FCONT_3:16

for X being set
for f being PartFunc of REAL,REAL st X c= dom f & f | X is monotone & f .: X = REAL holds
f | X is continuous

proof end;

theorem :: FCONT_3:17

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

proof end;

theorem :: FCONT_3:18

for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) & left_open_halfline p c= dom f holds
((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous

proof end;

theorem :: FCONT_3:19

for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | (right_open_halfline p) is increasing or f | (right_open_halfline p) is decreasing ) & right_open_halfline p c= dom f holds
((f | (right_open_halfline p)) ") | (f .: (right_open_halfline p)) is continuous

proof end;

theorem :: FCONT_3:20

for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | (left_closed_halfline p) is increasing or f | (left_closed_halfline p) is decreasing ) & left_closed_halfline p c= dom f holds
((f | (left_closed_halfline p)) ") | (f .: (left_closed_halfline p)) is continuous

proof end;

theorem :: FCONT_3:21

for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | (right_closed_halfline p) is increasing or f | (right_closed_halfline p) is decreasing ) & right_closed_halfline p c= dom f holds
((f | (right_closed_halfline p)) ") | (f .: (right_closed_halfline p)) is continuous

proof end;

theorem :: FCONT_3:22

for f being one-to-one PartFunc of REAL,REAL st ( f | ([#] REAL) is increasing or f | ([#] REAL) is decreasing ) & f is total holds
(f ") | (rng f) is continuous

proof end;

theorem :: FCONT_3:23

for g, p being Real
for f being PartFunc of REAL,REAL st ].p,g.[ c= dom f & f | ].p,g.[ is continuous & ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) holds
rng (f | ].p,g.[) is open

proof end;

theorem :: FCONT_3:24

for p being Real
for f being PartFunc of REAL,REAL st left_open_halfline p c= dom f & f | (left_open_halfline p) is continuous & ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) holds
rng (f | (left_open_halfline p)) is open

proof end;

theorem :: FCONT_3:25

for p being Real
for f being PartFunc of REAL,REAL st right_open_halfline p c= dom f & f | (right_open_halfline p) is continuous & ( f | (right_open_halfline p) is increasing or f | (right_open_halfline p) is decreasing ) holds
rng (f | (right_open_halfline p)) is open

proof end;

theorem :: FCONT_3:26

for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f | ([#] REAL) is continuous & ( f | ([#] REAL) is increasing or f | ([#] REAL) is decreasing ) holds
rng f is open

proof end;