Article FCONT_3, MML version 4.99.1005

:: FCONT_3:th 1
theorem
[#] REAL is closed;

:: FCONT_3:th 2
theorem
{} REAL is open;

:: FCONT_3:th 3
theorem
{} REAL is closed;

:: FCONT_3:th 4
theorem
[#] REAL is open;

:: FCONT_3:funcreg 1
registration
  cluster [#] REAL -> closed open;
end;

:: FCONT_3:funcreg 2
registration
  cluster {} REAL -> closed open;
end;

:: FCONT_3:th 5
theorem
for b1 being real set holds
   right_closed_halfline b1 is closed;

:: FCONT_3:th 6
theorem
for b1 being real set holds
   left_closed_halfline b1 is closed;

:: FCONT_3:th 7
theorem
for b1 being real set holds
   right_open_halfline b1 is open;

:: FCONT_3:th 8
theorem
for b1 being real set holds
   halfline b1 is open;

:: FCONT_3:funcreg 3
registration
  let a1 be real set;
  cluster right_open_halfline a1 -> open;
end;

:: FCONT_3:funcreg 4
registration
  let a1 be real set;
  cluster halfline a1 -> open;
end;

:: FCONT_3:funcreg 5
registration
  let a1 be real set;
  cluster right_closed_halfline a1 -> closed;
end;

:: FCONT_3:funcreg 6
registration
  let a1 be real set;
  cluster left_closed_halfline a1 -> closed;
end;

:: FCONT_3:th 9
theorem
for b1, b2 being Element of REAL
for b3 being real set holds
      0 < b3 & b1 in ].b2 - b3,b2 + b3.[
   iff
      ex b4 being Element of REAL st
         b1 = b2 + b4 & abs b4 < b3;

:: FCONT_3:th 10
theorem
for b1, b2 being Element of REAL
for b3 being real set holds
      0 < b3 & b1 in ].b2 - b3,b2 + b3.[
   iff
      b1 - b2 in ].- b3,b3.[;

:: FCONT_3:th 11
theorem
for b1 being Element of REAL holds
   left_closed_halfline b1 = {b1} \/ halfline b1;

:: FCONT_3:th 12
theorem
for b1 being Element of REAL holds
   right_closed_halfline b1 = {b1} \/ right_open_halfline b1;

:: FCONT_3:th 13
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being real set
      st for b4 being Element of NAT holds
           b2 . b4 = b3 - (b1 / (b4 + 1))
   holds b2 is convergent & lim b2 = b3;

:: FCONT_3:th 14
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being real set
      st for b4 being Element of NAT holds
           b2 . b4 = b3 + (b1 / (b4 + 1))
   holds b2 is convergent & lim b2 = b3;

:: FCONT_3:th 15
theorem
for b1 being Element of REAL
for b2 being real set
for b3 being Function-like Relation of REAL,REAL
      st b3 is_continuous_in b1 &
         b3 . b1 <> b2 &
         (ex b4 being Neighbourhood of b1 st
            b4 c= dom b3)
   holds ex b4 being Neighbourhood of b1 st
      b4 c= dom b3 &
       (for b5 being Element of REAL
             st b5 in b4
          holds b3 . b5 <> b2);

:: FCONT_3:th 16
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st (b2 is_increasing_on b1 or b2 is_decreasing_on b1)
   holds b2 | b1 is one-to-one;

:: FCONT_3:th 17
theorem
for b1 being set
for b2 being Function-like one-to-one Relation of REAL,REAL
      st b2 is_increasing_on b1
   holds (b2 | b1) " is_increasing_on b2 .: b1;

:: FCONT_3:th 18
theorem
for b1 being set
for b2 being Function-like one-to-one Relation of REAL,REAL
      st b2 is_decreasing_on b1
   holds (b2 | b1) " is_decreasing_on b2 .: b1;

:: FCONT_3:th 19
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         b2 is_monotone_on b1 &
         (ex b3 being Element of REAL st
            b2 .: b1 = halfline b3)
   holds b2 is_continuous_on b1;

:: FCONT_3:th 20
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         b2 is_monotone_on b1 &
         (ex b3 being Element of REAL st
            b2 .: b1 = right_open_halfline b3)
   holds b2 is_continuous_on b1;

:: FCONT_3:th 21
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         b2 is_monotone_on b1 &
         (ex b3 being Element of REAL st
            b2 .: b1 = left_closed_halfline b3)
   holds b2 is_continuous_on b1;

:: FCONT_3:th 22
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         b2 is_monotone_on b1 &
         (ex b3 being Element of REAL st
            b2 .: b1 = right_closed_halfline b3)
   holds b2 is_continuous_on b1;

:: FCONT_3:th 23
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         b2 is_monotone_on b1 &
         (ex b3, b4 being Element of REAL st
            b2 .: b1 = ].b3,b4.[)
   holds b2 is_continuous_on b1;

:: FCONT_3:th 24
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 & b2 is_monotone_on b1 & b2 .: b1 = REAL
   holds b2 is_continuous_on b1;

:: FCONT_3:th 25
theorem
for b1, b2 being Element of REAL
for b3 being Function-like one-to-one Relation of REAL,REAL
      st (b3 is_increasing_on ].b1,b2.[ or b3 is_decreasing_on ].b1,b2.[) &
         ].b1,b2.[ c= dom b3
   holds (b3 | ].b1,b2.[) " is_continuous_on b3 .: ].b1,b2.[;

:: FCONT_3:th 26
theorem
for b1 being Element of REAL
for b2 being Function-like one-to-one Relation of REAL,REAL
      st (b2 is_increasing_on halfline b1 or b2 is_decreasing_on halfline b1) & halfline b1 c= dom b2
   holds (b2 | halfline b1) " is_continuous_on b2 .: halfline b1;

:: FCONT_3:th 27
theorem
for b1 being Element of REAL
for b2 being Function-like one-to-one Relation of REAL,REAL
      st (b2 is_increasing_on right_open_halfline b1 or b2 is_decreasing_on right_open_halfline b1) &
         right_open_halfline b1 c= dom b2
   holds (b2 | right_open_halfline b1) " is_continuous_on b2 .: right_open_halfline b1;

:: FCONT_3:th 28
theorem
for b1 being Element of REAL
for b2 being Function-like one-to-one Relation of REAL,REAL
      st (b2 is_increasing_on left_closed_halfline b1 or b2 is_decreasing_on left_closed_halfline b1) &
         left_closed_halfline b1 c= dom b2
   holds (b2 | left_closed_halfline b1) " is_continuous_on b2 .: left_closed_halfline b1;

:: FCONT_3:th 29
theorem
for b1 being Element of REAL
for b2 being Function-like one-to-one Relation of REAL,REAL
      st (b2 is_increasing_on right_closed_halfline b1 or b2 is_decreasing_on right_closed_halfline b1) &
         right_closed_halfline b1 c= dom b2
   holds (b2 | right_closed_halfline b1) " is_continuous_on b2 .: right_closed_halfline b1;

:: FCONT_3:th 30
theorem
for b1 being Function-like one-to-one Relation of REAL,REAL
      st (b1 is_increasing_on [#] REAL or b1 is_decreasing_on [#] REAL) &
         b1 is total(REAL, REAL)
   holds b1 " is_continuous_on rng b1;

:: FCONT_3:th 31
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on ].b1,b2.[ &
         (b3 is_increasing_on ].b1,b2.[ or b3 is_decreasing_on ].b1,b2.[)
   holds rng (b3 | ].b1,b2.[) is open;

:: FCONT_3:th 32
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_continuous_on halfline b1 & (b2 is_increasing_on halfline b1 or b2 is_decreasing_on halfline b1)
   holds rng (b2 | halfline b1) is open;

:: FCONT_3:th 33
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_continuous_on right_open_halfline b1 &
         (b2 is_increasing_on right_open_halfline b1 or b2 is_decreasing_on right_open_halfline b1)
   holds rng (b2 | right_open_halfline b1) is open;

:: FCONT_3:th 34
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is_continuous_on [#] REAL &
         (b1 is_increasing_on [#] REAL or b1 is_decreasing_on [#] REAL)
   holds rng b1 is open;