This paper is concerned with the Cauchy Problem
\[
x
˙
(
t
)
=
f
(
t
,
x
(
t
)
)
,
x
(
t
0
)
=
x
0
∈
R
n
,
\dot x\left ( t \right ) = f\left ( {t,x\left ( t \right )} \right ),\quad x\left ( {{t_0}} \right ) = {x_0} \in {\mathbb {R}^n},
\]
where the vector field
f
f
may be discontinuous with respect to both variables
t
,
x
t,x
. If the total variation of
f
f
along certain directions is locally finite, we prove the existence of a unique solution, depending continuously on the initial data.