We consider the Walsh-Fourier series
∑
a
k
w
k
(
x
)
\sum {{a_k}{w_k}(x)}
of a function
f
f
assumed to be of bounded fluctuation on the interval
[
0
,
1
)
[0,1)
. Every function of bounded variation is also of bounded fluctuation on the same interval, but not conversely. We present an estimate for the difference of
f
(
x
)
f(x)
at a point
x
∈
[
0
,
1
)
x \in [0,1)
and the partial sum of its Walsh-Fourier series in terms of the bounded fluctuation operator. This gives rise to a local convergence result. As special cases, we obtain a Walsh analogue of the Dirichlet-Jordan test and a global convergence result due to Onneweer.