A method is given for constructing sample-continuous processes which do not satisfy the central limit theorem in
C
[
0
,
1
]
C[0,1]
. Let
{
X
(
t
)
:
t
∈
[
0
,
1
]
}
\{ X(t):t \in [0,1]\}
be a stochastic process. Using our method we characterize all possible nonnegative functions f for which the condition
\[
E
(
X
(
t
)
−
X
(
s
)
)
2
⩽
f
(
|
t
−
s
|
)
E( X(t) - X(s) )^2 \leqslant f( | t - s | )
\]
alone is sufficient to imply that
X
(
t
)
X(t)
satisfies the central limit theorem in
C
[
0
,
1
]
C[0,1]
.