We show that if
f
:
S
1
×
S
1
→
S
1
×
S
1
f \colon S^1 \times S^1 \to S^1 \times S^1
is
C
2
C^2
, with
f
(
x
,
t
)
=
(
f
t
(
x
)
,
t
)
f(x, t) = (f_t(x), t)
, and the rotation number of
f
t
f_t
is equal to
t
t
for all
t
∈
S
1
t \in S^1
, then
f
f
is topologically conjugate to the linear Dehn twist of the torus
(
1
a
m
p
;
1
0
a
m
p
;
1
)
\left ( \begin {smallmatrix} 1&1\\ 0&1 \end {smallmatrix} \right )
. We prove a differentiability result where the assumption that the rotation number of
f
t
f_t
is
t
t
is weakened to say that the rotation number is strictly monotone in
t
t
.