Article FCONT_1, MML version 4.99.1005
:: FCONT_1:prednot 1 => FCONT_1:pred 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
pred A1 is_continuous_in A2 means
a2 in dom a1 &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st rng b1 c= dom a1 & b1 is convergent & lim b1 = a2
holds a1 * b1 is convergent & a1 . a2 = lim (a1 * b1));
end;
:: FCONT_1:dfs 1
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
To prove
a1 is_continuous_in a2
it is sufficient to prove
thus a2 in dom a1 &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st rng b1 c= dom a1 & b1 is convergent & lim b1 = a2
holds a1 * b1 is convergent & a1 . a2 = lim (a1 * b1));
:: FCONT_1:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
b1 is_continuous_in b2
iff
b2 in dom b1 &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st rng b3 c= dom b1 & b3 is convergent & lim b3 = b2
holds b1 * b3 is convergent & b1 . b2 = lim (b1 * b3));
:: FCONT_1:th 2
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_continuous_in b1
iff
b1 in dom b2 &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st rng b3 c= dom b2 &
b3 is convergent &
lim b3 = b1 &
(for b4 being Element of NAT holds
b3 . b4 <> b1)
holds b2 * b3 is convergent & b2 . b1 = lim (b2 * b3));
:: FCONT_1:th 3
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_continuous_in b1
iff
b1 in dom b2 &
(for b3 being real set
st 0 < b3
holds ex b4 being real set st
0 < b4 &
(for b5 being real set
st b5 in dom b2 & abs (b5 - b1) < b4
holds abs ((b2 . b5) - (b2 . b1)) < b3));
:: FCONT_1:th 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
b1 is_continuous_in b2
iff
b2 in dom b1 &
(for b3 being Neighbourhood of b1 . b2 holds
ex b4 being Neighbourhood of b2 st
for b5 being real set
st b5 in dom b1 & b5 in b4
holds b1 . b5 in b3);
:: FCONT_1:th 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
b1 is_continuous_in b2
iff
b2 in dom b1 &
(for b3 being Neighbourhood of b1 . b2 holds
ex b4 being Neighbourhood of b2 st
b1 .: b4 c= b3);
:: FCONT_1:th 6
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
st b1 in dom b2 &
(ex b3 being Neighbourhood of b1 st
(dom b2) /\ b3 = {b1})
holds b2 is_continuous_in b1;
:: FCONT_1:th 7
theorem
for b1 being real set
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_continuous_in b1 & b3 is_continuous_in b1
holds b2 + b3 is_continuous_in b1 & b2 - b3 is_continuous_in b1 & b2 (#) b3 is_continuous_in b1;
:: FCONT_1:th 8
theorem
for b1, b2 being real set
for b3 being Function-like Relation of REAL,REAL
st b3 is_continuous_in b1
holds b2 (#) b3 is_continuous_in b1;
:: FCONT_1:th 9
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_in b1
holds abs b2 is_continuous_in b1 & - b2 is_continuous_in b1;
:: FCONT_1:th 10
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_in b1 & b2 . b1 <> 0
holds b2 ^ is_continuous_in b1;
:: FCONT_1:th 11
theorem
for b1 being real set
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_continuous_in b1 & b2 . b1 <> 0 & b3 is_continuous_in b1
holds b3 / b2 is_continuous_in b1;
:: FCONT_1:th 12
theorem
for b1 being real set
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_continuous_in b1 & b3 is_continuous_in b2 . b1
holds b3 * b2 is_continuous_in b1;
:: FCONT_1:prednot 2 => FCONT_1:pred 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_continuous_on A2 means
a2 c= dom a1 &
(for b1 being real set
st b1 in a2
holds a1 | a2 is_continuous_in b1);
end;
:: FCONT_1:dfs 2
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_continuous_on a2
it is sufficient to prove
thus a2 c= dom a1 &
(for b1 being real set
st b1 in a2
holds a1 | a2 is_continuous_in b1);
:: FCONT_1:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_continuous_on b2
iff
b2 c= dom b1 &
(for b3 being real set
st b3 in b2
holds b1 | b2 is_continuous_in b3);
:: FCONT_1:th 14
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_continuous_on b1
iff
b1 c= dom b2 &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st rng b3 c= b1 & b3 is convergent & lim b3 in b1
holds b2 * b3 is convergent & b2 . lim b3 = lim (b2 * b3));
:: FCONT_1:th 15
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_continuous_on b1
iff
b1 c= dom b2 &
(for b3, b4 being real set
st b3 in b1 & 0 < b4
holds ex b5 being real set st
0 < b5 &
(for b6 being real set
st b6 in b1 & abs (b6 - b3) < b5
holds abs ((b2 . b6) - (b2 . b3)) < b4));
:: FCONT_1:th 16
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_continuous_on b1
iff
b2 | b1 is_continuous_on b1;
:: FCONT_1:th 17
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
st b3 is_continuous_on b1 & b2 c= b1
holds b3 is_continuous_on b2;
:: FCONT_1:th 18
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
st b1 in dom b2
holds b2 is_continuous_on {b1};
:: FCONT_1:th 19
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1 & b3 is_continuous_on b1
holds b2 + b3 is_continuous_on b1 & b2 - b3 is_continuous_on b1 & b2 (#) b3 is_continuous_on b1;
:: FCONT_1:th 20
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
st b3 is_continuous_on b1 & b4 is_continuous_on b2
holds b3 + b4 is_continuous_on b1 /\ b2 & b3 - b4 is_continuous_on b1 /\ b2 & b3 (#) b4 is_continuous_on b1 /\ b2;
:: FCONT_1:th 21
theorem
for b1 being real set
for b2 being set
for b3 being Function-like Relation of REAL,REAL
st b3 is_continuous_on b2
holds b1 (#) b3 is_continuous_on b2;
:: FCONT_1:th 22
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1
holds abs b2 is_continuous_on b1 & - b2 is_continuous_on b1;
:: FCONT_1:th 23
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1 & b2 " {0} = {}
holds b2 ^ is_continuous_on b1;
:: FCONT_1:th 24
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1 &
(b2 | b1) " {0} = {}
holds b2 ^ is_continuous_on b1;
:: FCONT_1:th 25
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1 & b2 " {0} = {} & b3 is_continuous_on b1
holds b3 / b2 is_continuous_on b1;
:: FCONT_1:th 26
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1 & b3 is_continuous_on b2 .: b1
holds b3 * b2 is_continuous_on b1;
:: FCONT_1:th 27
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
st b3 is_continuous_on b1 & b4 is_continuous_on b2
holds b4 * b3 is_continuous_on b1 /\ (b3 " b2);
:: FCONT_1:th 28
theorem
for b1 being Function-like Relation of REAL,REAL
st b1 is total(REAL, REAL) &
(for b2, b3 being real set holds
b1 . (b2 + b3) = (b1 . b2) + (b1 . b3)) &
(ex b2 being real set st
b1 is_continuous_in b2)
holds b1 is_continuous_on REAL;
:: FCONT_1:th 29
theorem
for b1 being Function-like Relation of REAL,REAL
st dom b1 is compact & b1 is_continuous_on dom b1
holds rng b1 is compact;
:: FCONT_1:th 30
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_continuous_on b1
holds b2 .: b1 is compact;
:: FCONT_1:th 31
theorem
for b1 being Function-like Relation of REAL,REAL
st dom b1 <> {} & dom b1 is compact & b1 is_continuous_on dom b1
holds ex b2, b3 being real set st
b2 in dom b1 & b3 in dom b1 & b1 . b2 = upper_bound rng b1 & b1 . b3 = lower_bound rng b1;
:: FCONT_1:th 32
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_continuous_on b2
holds ex b3, b4 being real set st
b3 in b2 & b4 in b2 & b1 . b3 = upper_bound (b1 .: b2) & b1 . b4 = lower_bound (b1 .: b2);
:: FCONT_1:prednot 3 => FCONT_1:pred 3
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_Lipschitzian_on A2 means
a2 c= dom a1 &
(ex b1 being real set st
0 < b1 &
(for b2, b3 being real set
st b2 in a2 & b3 in a2
holds abs ((a1 . b2) - (a1 . b3)) <= b1 * abs (b2 - b3)));
end;
:: FCONT_1:dfs 3
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_Lipschitzian_on a2
it is sufficient to prove
thus a2 c= dom a1 &
(ex b1 being real set st
0 < b1 &
(for b2, b3 being real set
st b2 in a2 & b3 in a2
holds abs ((a1 . b2) - (a1 . b3)) <= b1 * abs (b2 - b3)));
:: FCONT_1:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_Lipschitzian_on b2
iff
b2 c= dom b1 &
(ex b3 being real set st
0 < b3 &
(for b4, b5 being real set
st b4 in b2 & b5 in b2
holds abs ((b1 . b4) - (b1 . b5)) <= b3 * abs (b4 - b5)));
:: FCONT_1:th 34
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
st b3 is_Lipschitzian_on b1 & b2 c= b1
holds b3 is_Lipschitzian_on b2;
:: FCONT_1:th 35
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
st b3 is_Lipschitzian_on b1 & b4 is_Lipschitzian_on b2
holds b3 + b4 is_Lipschitzian_on b1 /\ b2;
:: FCONT_1:th 36
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
st b3 is_Lipschitzian_on b1 & b4 is_Lipschitzian_on b2
holds b3 - b4 is_Lipschitzian_on b1 /\ b2;
:: FCONT_1:th 37
theorem
for b1, b2, b3, b4 being set
for b5, b6 being Function-like Relation of REAL,REAL
st b5 is_Lipschitzian_on b1 & b6 is_Lipschitzian_on b2 & b5 is_bounded_on b3 & b6 is_bounded_on b4
holds b5 (#) b6 is_Lipschitzian_on ((b1 /\ b3) /\ b2) /\ b4;
:: FCONT_1:th 38
theorem
for b1 being set
for b2 being real set
for b3 being Function-like Relation of REAL,REAL
st b3 is_Lipschitzian_on b1
holds b2 (#) b3 is_Lipschitzian_on b1;
:: FCONT_1:th 39
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_Lipschitzian_on b1
holds - b2 is_Lipschitzian_on b1 & abs b2 is_Lipschitzian_on b1;
:: FCONT_1:th 40
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_Lipschitzian_on b1;
:: FCONT_1:th 41
theorem
for b1 being Element of bool REAL holds
id b1 is_Lipschitzian_on b1;
:: FCONT_1:th 42
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_Lipschitzian_on b1
holds b2 is_continuous_on b1;
:: FCONT_1:th 43
theorem
for b1 being Function-like Relation of REAL,REAL
st ex b2 being real set st
rng b1 = {b2}
holds b1 is_continuous_on dom b1;
:: FCONT_1:th 44
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_continuous_on b1;
:: FCONT_1:th 45
theorem
for b1 being Function-like Relation of REAL,REAL
st for b2 being real set
st b2 in dom b1
holds b1 . b2 = b2
holds b1 is_continuous_on dom b1;
:: FCONT_1:th 46
theorem
for b1 being Function-like Relation of REAL,REAL
st b1 = id dom b1
holds b1 is_continuous_on dom b1;
:: FCONT_1:th 47
theorem
for b1 being Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 & b2 | b1 = id b1
holds b2 is_continuous_on b1;
:: FCONT_1:th 48
theorem
for b1 being set
for b2, b3 being real set
for b4 being Function-like Relation of REAL,REAL
st b1 c= dom b4 &
(for b5 being real set
st b5 in b1
holds b4 . b5 = (b2 * b5) + b3)
holds b4 is_continuous_on b1;
:: FCONT_1:th 49
theorem
for b1 being Function-like Relation of REAL,REAL
st for b2 being real set
st b2 in dom b1
holds b1 . b2 = b2 ^2
holds b1 is_continuous_on dom b1;
:: FCONT_1:th 50
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 &
(for b3 being real set
st b3 in b1
holds b2 . b3 = b3 ^2)
holds b2 is_continuous_on b1;
:: FCONT_1:th 51
theorem
for b1 being Function-like Relation of REAL,REAL
st for b2 being real set
st b2 in dom b1
holds b1 . b2 = abs b2
holds b1 is_continuous_on dom b1;
:: FCONT_1:th 52
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 &
(for b3 being real set
st b3 in b1
holds b2 . b3 = abs b3)
holds b2 is_continuous_on b1;
:: FCONT_1:th 53
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 real set st
b3 <= b4 & b2 .: b1 = [.b3,b4.])
holds b2 is_continuous_on b1;
:: FCONT_1:th 54
theorem
for b1, b2 being real set
for b3 being Function-like one-to-one Relation of REAL,REAL
st b1 <= b2 &
[.b1,b2.] c= dom b3 &
(b3 is_increasing_on [.b1,b2.] or b3 is_decreasing_on [.b1,b2.])
holds (b3 | [.b1,b2.]) " is_continuous_on b3 .: [.b1,b2.];