Let
C
(
X
,
E
)
C(X,E)
be the linear space of all continuous functions on a compact Hausdorff space
X
X
with values in a locally convex space
E
E
. We characterize maps
T
:
C
(
X
,
E
)
→
C
(
Y
,
E
)
T:C(X,E)\to C(Y,E)
which satisfy
R
a
n
(
T
F
−
T
G
)
⊂
R
a
n
(
F
−
G
)
\mathrm {Ran}(TF-TG)\subset \mathrm {Ran}(F-G)
for all
F
,
G
∈
C
(
X
,
E
)
F,G\in C(X,E)
.