Ü
f, g : X R h(x) := max{f(x), g(x)}.
h
a " R
{x : h(x) < a} =
= {x : max{f(x), g(x)} < a} =
= {x : g(x) < a '" f(x) < a} =
= {x : g(x) < a} )" {x : f(x) < a}
f g Ã- {x : h(x) < a} " S
h
Ü
A ‚" R f : R R
x x " A
f(x) = .
-x x " A
f
f g(x) = x x " R
2x x " A
f+g f(x)+g(x) =
0 x " A
{x " R : f(x) + g(x) > 0} = A )" (0, ")
{x " R : f(x) + g(x) < 0} = A )" (-", 0)
A )" (0, ") A )" (-", 0)
A\{0} = [A )" (0, ")] *" [A )" (-", 0)],
A [A\{0}] = A [A\{0}] *" {0} = A
f + g
(1) =Ò! (2)
(2) =Ò! (1) a " R qn
qn d" a qn a
{x " X : f(x) < a} = {x " X : f(x) < qn} (")
n"N
‚" x " X f(x) < a n f(x) < qn d" a
qn a x " {y : f(y) < qn} ‚" {y " Y : f(y) < qn}.
n"N
ƒ" x " {y " Y : f(y) < qn} n " N
n"N
f(x) < qn d" a x " {y " X : f(y) < a}
(")
{x " X : f(x) < qn} " S
{x : f(x) < a} " S S Ã-
f(x) := x x " R f
Ã
a " R {f(x) < a} =
k " Z k < a d" k +1 {x " R : f(x) < a} = (-", k +1) "
S.
f
S = {A : A- R\A- } f
{x : f(x) < 0} = (-", 0) " S
(1) =Ò! (2)
U U = (an, bn) an, bn.
n
f-1(U) = f-1( (an, bn)) = f-1((an, bn)) = {x : an < f(x) < bn} =
n n n
({x : an < f(x)} )" {x : f(x) < bn})
n
 Ã- f f-1(U) " S.
(2) =Ò! (3)
F
f-1(F ) = R\f-1(R\F )
R \ F Ã- f-1(F ) " S.
(3) =Ò! (1)
a " R
{x : f(x) < a} = R\{x : f(x) e" a} = R\f-1([a, "))
f : R R
Ã- {x : f(x) < a} " S.
" x " E
f(x) := S - Ã- X, E ‚" X
-" x " E
f U ‚" R
f-1(U) " S.
f {x : f(x) < 0} = E " S,
U ‚" R f-1(U) = {x : f(x) " U} = " " S