Let
R
\mathfrak {R}
be an o-minimal expansion of a divisible ordered abelian group
(
R
,
>
,
+
,
0
,
1
)
(R,>,+,0,1)
with a distinguished positive element
1
1
. Then the following dichotomy holds: Either there is a
0
0
-definable binary operation
⋅
\cdot
such that
(
R
,
>
,
+
,
⋅
,
0
,
1
)
(R,>,+,\cdot ,0,1)
is an ordered real closed field; or, for every definable function
f
:
R
→
R
f:R\to R
there exists a
0
0
-definable
λ
∈
{
0
}
∪
Aut
(
R
,
+
)
\lambda \in \{0\}\cup \operatorname {Aut}(R,+)
with
lim
x
→
+
∞
[
f
(
x
)
−
λ
(
x
)
]
∈
R
\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R
. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure
M
:=
(
M
,
>
,
…
)
\mathfrak {M}:=(M,>,\dots )
there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order)
M
\mathfrak {M}
-definable groups with underlying set
M
M
.