In this paper, parabolic operators of the form
\[
u
t
−
div
A
(
x
,
t
,
u
,
D
u
)
−
B
(
x
,
t
,
u
,
D
u
)
{u_t} - \operatorname {div} A(x,\,t,\,u,\,Du) - B(x,\,t,\,u,\,Du)
\]
are considered where
A
A
and
B
B
are Borel measurable and subject to linear growth conditions. Let
ψ
:
Ω
→
R
1
\psi :\,\Omega \to {R^1}
be a Borel function bounded above (an obstacle) where
Ω
⊂
R
n
+
1
\Omega \subset {R^{n + 1}}
. Let
u
∈
W
1
,
2
(
Ω
)
u \in {W^{1,2}}(\Omega )
be a weak solution of the variational inequality in the following sense: assume that
u
⩾
ψ
u \geqslant \psi
q.e. and
\[
∫
Ω
u
t
φ
+
A
⋅
D
φ
−
B
φ
⩾
0
\int _\Omega {{u_t}\varphi + A \cdot D\varphi - B\varphi \geqslant 0}
\]
whenever
φ
∈
W
0
1
,
2
(
Ω
)
\varphi \in W_0^{1,2}(\Omega )
and
φ
⩾
u
−
ψ
\varphi \geqslant u - \psi
q.e. Here q.e. means everywhere except for a set of classical parabolic capacity. It is shown that
u
u
is continuous even though the obstacle may be discontinuous. A mild condition on
ψ
\psi
which can be expressed in terms of the fine topology is sufficient to ensure the continuity of
u
u
. A modulus of continuity is obtained for
u
u
in terms of the data given for
ψ
\psi
.