Definitions and integrals are of the subdivision-refinement type, and functions are from
R
×
R
R \times R
to R, where R represents the real numbers. Let
O
M
∘
O{M^ \circ }
be the class of functions G such that
x
∏
y
(
1
+
G
)
_x\prod {^y} (1 + G)
exists for
a
≦
x
>
y
≦
b
a \leqq x > y \leqq b
and
∫
a
b
|
1
+
G
−
∏
(
1
+
G
)
|
=
0
\smallint _a^b|1 + G - \prod {(1 + G)| = 0}
. Let
O
P
∘
O{P^ \circ }
be the class of functions G such that
|
∏
q
=
i
j
(
1
+
G
q
)
|
|\prod \nolimits _{q = i}^j {(1 + {G_q})|}
is bounded for refinements
{
x
q
}
q
=
0
n
\{ {x_q}\} _{q = 0}^n
of a suitable subdivision of [a, b]. If F and G are functions from
R
×
R
R \times R
to R such that
F
∈
O
P
∘
F \in O{P^ \circ }
on [a, b],
lim
x
,
y
→
p
+
F
(
x
,
y
)
{\lim _{x,y \to {p^ + }}}F(x,y)
and
lim
x
,
y
→
p
−
F
(
x
,
y
)
{\lim _{x,y \to {p^ - }}}F(x,y)
exist and are zero for
p
∈
[
a
,
b
]
p \in [a,b]
, each of
lim
x
→
p
+
F
(
p
,
x
)
,
lim
x
→
p
−
F
(
x
,
p
)
,
lim
x
→
p
+
G
(
p
,
x
)
{\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)
and
lim
x
→
p
−
G
(
x
,
p
)
{\lim _{x \to {p^ - }}}G(x,p)
exist for
p
∈
[
a
,
b
]
p \in [a,b]
, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1)
F
+
G
∈
O
M
∘
F + G \in O{M^ \circ }
on [a, b], (2)
F
∈
O
M
∘
F \in O{M^ \circ }
on [a, b], and (3)
G
∈
O
M
∘
G \in O{M^\circ }
on [a, b].