Surjective isometries between some classical function spaces are investigated. We give a simple technical scheme which verifies whether any such isometry is given by a homeomorphism between corresponding Hausdorff compact spaces. In particular the answer is positive for the
C
1
(
X
)
{C^1}(X)
,
AC
[
0
,
1
]
\operatorname {AC} [0,1]
,
Lip
α
(
X
)
{\operatorname {Lip} _\alpha }(X)
and
lip
α
(
X
)
{\operatorname {lip} _\alpha }(X)
spaces provided with various natural norms.