let X be set ; :: thesis: for f being PartFunc of REAL,REAL st X c= dom f & f | X is monotone & f .: X = REAL holds
f | X is continuous
let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & f | X is monotone & f .: X = REAL implies f | X is continuous )
assume that
A1: X c= dom f and
A2: f | X is monotone and
A3: f .: X = REAL ; :: thesis: f | X is continuous

now :: thesis: f | X is continuous

per cases ( f | X is non-decreasing or f | X is non-increasing ) by A2, RFUNCT_2:def 5;

suppose f | X is non-decreasing ; :: thesis: f | X is continuous

then A4: (f | X) | X is non-decreasing ;
for x0 being real number st x0 in dom (f | X) holds
f | X is_continuous_in x0

proof

let x0 be real number ; :: thesis: ( x0 in dom (f | X) implies f | X is_continuous_in x0 )
A5: x0 in REAL by XREAL_0:def 1;
A6: (f | X) .: X = f .: X by RELAT_1:129;
assume x0 in dom (f | X) ; :: thesis: f | X is_continuous_in x0
then x0 in X ;
then x0 in (dom f) /\ X by A1, XBOOLE_0:def 4;
then A7: x0 in dom (f | X) by RELAT_1:61;

now :: thesis: for N1 being Neighbourhood of (f | X) . x0 ex N being Neighbourhood of x0 st
for x being real number st x in dom (f | X) & x in N holds
(f | X) . x in N1

let N1 be Neighbourhood of (f | X) . x0; :: thesis: ex N being Neighbourhood of x0 st
for x being real number st x in dom (f | X) & x in N holds
(f | X) . x in N1
consider r being real number such that
A8: r > 0 and
A9: N1 = ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by RCOMP_1:def 6;
reconsider r = r as Real by XREAL_0:def 1;
A10: (f | X) . x0 < ((f | X) . x0) + (r / 2) by A8, XREAL_1:29, XREAL_1:215;
set M1 = ((f | X) . x0) - (r / 2);
consider r1 being Real such that
A11: ( r1 in dom (f | X) & r1 in X ) and
A12: ((f | X) . x0) - (r / 2) = (f | X) . r1 by A3, A6, PARTFUN2:59;
A13: (f | X) . x0 < ((f | X) . x0) + (r / 2) by A8, XREAL_1:29, XREAL_1:215;
then A14: ((f | X) . x0) - (r / 2) < (f | X) . x0 by XREAL_1:19;

A15: now :: thesis: not x0 < r1

assume A16: x0 < r1 ; :: thesis: contradiction
( x0 in X /\ (dom (f | X)) & r1 in X /\ (dom (f | X)) ) by A7, A11, XBOOLE_0:def 4;
hence contradiction by A4, A12, A14, A16, RFUNCT_2:22, A5; :: thesis: verum

end;

set M2 = ((f | X) . x0) + (r / 2);
consider r2 being Real such that
A17: ( r2 in dom (f | X) & r2 in X ) and
A18: ((f | X) . x0) + (r / 2) = (f | X) . r2 by A3, A6, PARTFUN2:59;
A19: ((f | X) . x0) + (r / 2) > (f | X) . x0 by A8, XREAL_1:29, XREAL_1:215;

A20: now :: thesis: not r2 < x0

assume A21: r2 < x0 ; :: thesis: contradiction
( x0 in X /\ (dom (f | X)) & r2 in X /\ (dom (f | X)) ) by A7, A17, XBOOLE_0:def 4;
hence contradiction by A4, A18, A19, A21, RFUNCT_2:22, A5; :: thesis: verum

end;

x0 <> r2 by A8, A18, XREAL_1:29, XREAL_1:215;
then x0 < r2 by A20, XXREAL_0:1;
then A22: r2 - x0 > 0 by XREAL_1:50;
set R = min ((x0 - r1),(r2 - x0));
A23: min ((x0 - r1),(r2 - x0)) <= r2 - x0 by XXREAL_0:17;
r1 <> x0 by A12, A13, XREAL_1:19;
then r1 < x0 by A15, XXREAL_0:1;
then x0 - r1 > 0 by XREAL_1:50;
then min ((x0 - r1),(r2 - x0)) > 0 by A22, XXREAL_0:15;
then reconsider N = ].(x0 - (min ((x0 - r1),(r2 - x0)))),(x0 + (min ((x0 - r1),(r2 - x0)))).[ as Neighbourhood of x0 by RCOMP_1:def 6;
take N = N; :: thesis: for x being real number st x in dom (f | X) & x in N holds
(f | X) . x in N1
let x be real number ; :: thesis: ( x in dom (f | X) & x in N implies (f | X) . x in N1 )
A24: x in REAL by XREAL_0:def 1;
assume that
A25: x in dom (f | X) and
A26: x in N ; :: thesis: (f | X) . x in N1
A27: x in X /\ (dom (f | X)) by A25, XBOOLE_1:28;
A28: ((f | X) . x0) + (r / 2) < (((f | X) . x0) + (r / 2)) + (r / 2) by A8, XREAL_1:29, XREAL_1:215;
A29: (f | X) . x0 < ((f | X) . x0) + r by A8, XREAL_1:29;
then ((f | X) . x0) - r < (((f | X) . x0) + r) - r by XREAL_1:9;
then ((f | X) . x0) - r < ((f | X) . x0) + (r / 2) by A10, XXREAL_0:2;
then A30: ((f | X) . x0) + (r / 2) in ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by A28;
A31: ((f | X) . x0) - r < (((f | X) . x0) - r) + (r / 2) by A8, XREAL_1:29, XREAL_1:215;
((f | X) . x0) - (r / 2) < (f | X) . x0 by A10, XREAL_1:19;
then ((f | X) . x0) - (r / 2) < ((f | X) . x0) + r by A29, XXREAL_0:2;
then ((f | X) . x0) - (r / 2) in ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by A31;
then A32: [.(((f | X) . x0) - (r / 2)),(((f | X) . x0) + (r / 2)).] c= ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by A30, XXREAL_2:def 12;
A33: ex s being Real st
( s = x & x0 - (min ((x0 - r1),(r2 - x0))) < s & s < x0 + (min ((x0 - r1),(r2 - x0))) ) by A26;
then x0 < (min ((x0 - r1),(r2 - x0))) + x by XREAL_1:19;
then A34: x0 - x < ((min ((x0 - r1),(r2 - x0))) + x) - x by XREAL_1:9;
min ((x0 - r1),(r2 - x0)) <= x0 - r1 by XXREAL_0:17;
then x0 - x < x0 - r1 by A34, XXREAL_0:2;
then - (x0 - x) > - (x0 - r1) by XREAL_1:24;
then A35: (x - x0) + x0 > (r1 - x0) + x0 by XREAL_1:6;
r1 in X /\ (dom (f | X)) by A11, XBOOLE_0:def 4;
then A36: (f | X) . r1 <= (f | X) . x by A4, A35, A27, RFUNCT_2:22, A24;
x - x0 < min ((x0 - r1),(r2 - x0)) by A33, XREAL_1:19;
then x - x0 < r2 - x0 by A23, XXREAL_0:2;
then A37: (x - x0) + x0 < (r2 - x0) + x0 by XREAL_1:6;
r2 in X /\ (dom (f | X)) by A17, XBOOLE_0:def 4;
then (f | X) . x <= (f | X) . r2 by A4, A37, A27, RFUNCT_2:22, A24;
then (f | X) . x in [.(((f | X) . x0) - (r / 2)),(((f | X) . x0) + (r / 2)).] by A12, A18, A36;
hence (f | X) . x in N1 by A9, A32; :: thesis: verum

end;

hence f | X is_continuous_in x0 by FCONT_1:4; :: thesis: verum

end;

hence f | X is continuous by FCONT_1:def 2; :: thesis: verum

end;

suppose f | X is non-increasing ; :: thesis: f | X is continuous

then A38: (f | X) | X is non-increasing ;
for x0 being real number st x0 in dom (f | X) holds
f | X is_continuous_in x0

proof

let x0 be real number ; :: thesis: ( x0 in dom (f | X) implies f | X is_continuous_in x0 )
A39: x0 in REAL by XREAL_0:def 1;
A40: (f | X) .: X = f .: X by RELAT_1:129;
assume x0 in dom (f | X) ; :: thesis: f | X is_continuous_in x0
then x0 in X ;
then x0 in (dom f) /\ X by A1, XBOOLE_0:def 4;
then A41: x0 in dom (f | X) by RELAT_1:61;

now :: thesis: for N1 being Neighbourhood of (f | X) . x0 ex N being Neighbourhood of x0 st
for x being real number st x in dom (f | X) & x in N holds
(f | X) . x in N1

let N1 be Neighbourhood of (f | X) . x0; :: thesis: ex N being Neighbourhood of x0 st
for x being real number st x in dom (f | X) & x in N holds
(f | X) . x in N1
consider r being real number such that
A42: r > 0 and
A43: N1 = ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by RCOMP_1:def 6;
reconsider r = r as Real by XREAL_0:def 1;
A44: (f | X) . x0 < ((f | X) . x0) + (r / 2) by A42, XREAL_1:29, XREAL_1:215;
set M1 = ((f | X) . x0) - (r / 2);
consider r1 being Real such that
A45: ( r1 in dom (f | X) & r1 in X ) and
A46: ((f | X) . x0) - (r / 2) = (f | X) . r1 by A3, A40, PARTFUN2:59;
A47: (f | X) . x0 < ((f | X) . x0) + (r / 2) by A42, XREAL_1:29, XREAL_1:215;
then A48: ((f | X) . x0) - (r / 2) < (f | X) . x0 by XREAL_1:19;

A49: now :: thesis: not x0 > r1

assume A50: x0 > r1 ; :: thesis: contradiction
( x0 in X /\ (dom (f | X)) & r1 in X /\ (dom (f | X)) ) by A41, A45, XBOOLE_0:def 4;
hence contradiction by A38, A46, A48, A50, RFUNCT_2:23, A39; :: thesis: verum

end;

set M2 = ((f | X) . x0) + (r / 2);
consider r2 being Real such that
A51: ( r2 in dom (f | X) & r2 in X ) and
A52: ((f | X) . x0) + (r / 2) = (f | X) . r2 by A3, A40, PARTFUN2:59;
A53: ((f | X) . x0) + (r / 2) > (f | X) . x0 by A42, XREAL_1:29, XREAL_1:215;

A54: now :: thesis: not r2 > x0

assume A55: r2 > x0 ; :: thesis: contradiction
( x0 in X /\ (dom (f | X)) & r2 in X /\ (dom (f | X)) ) by A41, A51, XBOOLE_0:def 4;
hence contradiction by A38, A52, A53, A55, RFUNCT_2:23, A39; :: thesis: verum

end;

x0 <> r2 by A42, A52, XREAL_1:29, XREAL_1:215;
then x0 > r2 by A54, XXREAL_0:1;
then A56: x0 - r2 > 0 by XREAL_1:50;
set R = min ((r1 - x0),(x0 - r2));
A57: min ((r1 - x0),(x0 - r2)) <= r1 - x0 by XXREAL_0:17;
r1 <> x0 by A46, A47, XREAL_1:19;
then r1 > x0 by A49, XXREAL_0:1;
then r1 - x0 > 0 by XREAL_1:50;
then min ((r1 - x0),(x0 - r2)) > 0 by A56, XXREAL_0:15;
then reconsider N = ].(x0 - (min ((r1 - x0),(x0 - r2)))),(x0 + (min ((r1 - x0),(x0 - r2)))).[ as Neighbourhood of x0 by RCOMP_1:def 6;
take N = N; :: thesis: for x being real number st x in dom (f | X) & x in N holds
(f | X) . x in N1
let x be real number ; :: thesis: ( x in dom (f | X) & x in N implies (f | X) . x in N1 )
A58: x in REAL by XREAL_0:def 1;
assume that
A59: x in dom (f | X) and
A60: x in N ; :: thesis: (f | X) . x in N1
A61: x in X /\ (dom (f | X)) by A59, XBOOLE_1:28;
A62: ((f | X) . x0) + (r / 2) < (((f | X) . x0) + (r / 2)) + (r / 2) by A42, XREAL_1:29, XREAL_1:215;
A63: (f | X) . x0 < ((f | X) . x0) + r by A42, XREAL_1:29;
then ((f | X) . x0) - r < (((f | X) . x0) + r) - r by XREAL_1:9;
then ((f | X) . x0) - r < ((f | X) . x0) + (r / 2) by A44, XXREAL_0:2;
then A64: ((f | X) . x0) + (r / 2) in ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by A62;
A65: ((f | X) . x0) - r < (((f | X) . x0) - r) + (r / 2) by A42, XREAL_1:29, XREAL_1:215;
((f | X) . x0) - (r / 2) < (f | X) . x0 by A44, XREAL_1:19;
then ((f | X) . x0) - (r / 2) < ((f | X) . x0) + r by A63, XXREAL_0:2;
then ((f | X) . x0) - (r / 2) in ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by A65;
then A66: [.(((f | X) . x0) - (r / 2)),(((f | X) . x0) + (r / 2)).] c= ].(((f | X) . x0) - r),(((f | X) . x0) + r).[ by A64, XXREAL_2:def 12;
A67: ex s being Real st
( s = x & x0 - (min ((r1 - x0),(x0 - r2))) < s & s < x0 + (min ((r1 - x0),(x0 - r2))) ) by A60;
then x0 < (min ((r1 - x0),(x0 - r2))) + x by XREAL_1:19;
then A68: x0 - x < ((min ((r1 - x0),(x0 - r2))) + x) - x by XREAL_1:9;
x - x0 < min ((r1 - x0),(x0 - r2)) by A67, XREAL_1:19;
then x - x0 < r1 - x0 by A57, XXREAL_0:2;
then A69: (x - x0) + x0 < (r1 - x0) + x0 by XREAL_1:6;
r1 in X /\ (dom (f | X)) by A45, XBOOLE_0:def 4;
then A70: (f | X) . r1 <= (f | X) . x by A38, A69, A61, RFUNCT_2:23, A58;
min ((r1 - x0),(x0 - r2)) <= x0 - r2 by XXREAL_0:17;
then x0 - x < x0 - r2 by A68, XXREAL_0:2;
then - (x0 - x) > - (x0 - r2) by XREAL_1:24;
then A71: (x - x0) + x0 > (r2 - x0) + x0 by XREAL_1:6;
r2 in X /\ (dom (f | X)) by A51, XBOOLE_0:def 4;
then (f | X) . x <= (f | X) . r2 by A38, A71, A61, RFUNCT_2:23, A58;
then (f | X) . x in [.(((f | X) . x0) - (r / 2)),(((f | X) . x0) + (r / 2)).] by A46, A52, A70;
hence (f | X) . x in N1 by A43, A66; :: thesis: verum

end;

hence f | X is_continuous_in x0 by FCONT_1:4; :: thesis: verum

end;

hence f | X is continuous by FCONT_1:def 2; :: thesis: verum

end;

end;

end;

hence f | X is continuous ; :: thesis: verum