:: Monotonic and Continuous Real Function
:: by Jaros{\l}aw Kotowicz
::
:: Copyright (c) 1991-2017 Association of Mizar Users

registration
coherence 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
coherence
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;
coherence
proof end;
coherence
proof end;
coherence
proof end;
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 | is increasing or f | is decreasing ) & left_open_halfline p c= dom f holds
((f | ) ") | (f .: ) is continuous
proof end;

theorem :: FCONT_3:19
for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | is increasing or f | is decreasing ) & right_open_halfline p c= dom f holds
((f | ) ") | () is continuous
proof end;

theorem :: FCONT_3:20
for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | is increasing or f | is decreasing ) & left_closed_halfline p c= dom f holds
(() ") | () is continuous
proof end;

theorem :: FCONT_3:21
for p being Real
for f being one-to-one PartFunc of REAL,REAL st ( f | is increasing or f | is decreasing ) & right_closed_halfline p c= dom f holds
(() ") | () is continuous
proof end;

theorem :: FCONT_3:22
for f being one-to-one PartFunc of REAL,REAL st ( f | () is increasing or f | () 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 | is continuous & ( f | is increasing or f | is decreasing ) holds
rng (f | ) 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 | is continuous & ( f | is increasing or f | is decreasing ) holds
rng (f | ) is open
proof end;

theorem :: FCONT_3:26
for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f | () is continuous & ( f | () is increasing or f | () is decreasing ) holds
rng f is open
proof end;