Article FCONT_2, MML version 4.99.1005
:: FCONT_2:prednot 1 => FCONT_2:pred 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_uniformly_continuous_on A2 means
a2 c= dom a1 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of REAL
st b3 in a2 & b4 in a2 & abs (b3 - b4) < b2
holds abs ((a1 . b3) - (a1 . b4)) < b1));
end;
:: FCONT_2:dfs 1
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_uniformly_continuous_on a2
it is sufficient to prove
thus a2 c= dom a1 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of REAL
st b3 in a2 & b4 in a2 & abs (b3 - b4) < b2
holds abs ((a1 . b3) - (a1 . b4)) < b1));
:: FCONT_2:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_uniformly_continuous_on b2
iff
b2 c= dom b1 &
(for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of REAL st
0 < b4 &
(for b5, b6 being Element of REAL
st b5 in b2 & b6 in b2 & abs (b5 - b6) < b4
holds abs ((b1 . b5) - (b1 . b6)) < b3));
:: FCONT_2:th 2
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
st b3 is_uniformly_continuous_on b1 & b2 c= b1
holds b3 is_uniformly_continuous_on b2;
:: FCONT_2:th 3
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
st b3 is_uniformly_continuous_on b1 & b4 is_uniformly_continuous_on b2
holds b3 + b4 is_uniformly_continuous_on b1 /\ b2;
:: FCONT_2:th 4
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
st b3 is_uniformly_continuous_on b1 & b4 is_uniformly_continuous_on b2
holds b3 - b4 is_uniformly_continuous_on b1 /\ b2;
:: FCONT_2:th 5
theorem
for b1 being set
for b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_uniformly_continuous_on b1
holds b2 (#) b3 is_uniformly_continuous_on b1;
:: FCONT_2:th 6
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_uniformly_continuous_on b1
holds - b2 is_uniformly_continuous_on b1;
:: FCONT_2:th 7
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_uniformly_continuous_on b1
holds abs b2 is_uniformly_continuous_on b1;
:: FCONT_2:th 8
theorem
for b1, b2, b3, b4 being set
for b5, b6 being Function-like Relation of REAL,REAL
st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2 & b5 is_bounded_on b3 & b6 is_bounded_on b4
holds b5 (#) b6 is_uniformly_continuous_on ((b1 /\ b3) /\ b2) /\ b4;
:: FCONT_2:th 9
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_uniformly_continuous_on b1
holds b2 is_continuous_on b1;
:: FCONT_2:th 10
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_Lipschitzian_on b1
holds b2 is_uniformly_continuous_on b1;
:: FCONT_2:th 11
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
st b2 is compact & b1 is_continuous_on b2
holds b1 is_uniformly_continuous_on b2;
:: FCONT_2:th 13
theorem
for b1 being Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 & b1 is compact & b2 is_uniformly_continuous_on b1
holds b2 .: b1 is compact;
:: FCONT_2:th 14
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
st b2 <> {} & b2 c= dom b1 & b2 is compact & b1 is_uniformly_continuous_on b2
holds ex b3, b4 being Element of REAL st
b3 in b2 & b4 in b2 & b1 . b3 = upper_bound (b1 .: b2) & b1 . b4 = lower_bound (b1 .: b2);
:: FCONT_2:th 15
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 & b2 is_constant_on b1
holds b2 is_uniformly_continuous_on b1;
:: FCONT_2:th 16
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
for b4 being Element of REAL
st b4 in [.b3 . b1,b3 . b2.] \/ [.b3 . b2,b3 . b1.]
holds ex b5 being Element of REAL st
b5 in [.b1,b2.] & b4 = b3 . b5;
:: FCONT_2:th 17
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
for b4 being Element of REAL
st b4 in [.lower_bound (b3 .: [.b1,b2.]),upper_bound (b3 .: [.b1,b2.]).]
holds ex b5 being Element of REAL st
b5 in [.b1,b2.] & b4 = b3 . b5;
:: FCONT_2:th 18
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is one-to-one & b1 <= b2 & b3 is_continuous_on [.b1,b2.] & not b3 is_increasing_on [.b1,b2.]
holds b3 is_decreasing_on [.b1,b2.];
:: FCONT_2:th 19
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is one-to-one &
b1 <= b2 &
b3 is_continuous_on [.b1,b2.] &
(lower_bound (b3 .: [.b1,b2.]) = b3 . b1 implies upper_bound (b3 .: [.b1,b2.]) <> b3 . b2)
holds lower_bound (b3 .: [.b1,b2.]) = b3 . b2 &
upper_bound (b3 .: [.b1,b2.]) = b3 . b1;
:: FCONT_2:th 20
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
holds b3 .: [.b1,b2.] = [.lower_bound (b3 .: [.b1,b2.]),upper_bound (b3 .: [.b1,b2.]).];
:: FCONT_2:th 21
theorem
for b1, b2 being Element of REAL
for b3 being Function-like one-to-one Relation of REAL,REAL
st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
holds b3 " is_continuous_on [.lower_bound (b3 .: [.b1,b2.]),upper_bound (b3 .: [.b1,b2.]).];