where a e R d, f and gi (i = 1 . . . . . k) are vector fields on R ~, and z 1 . . . . . z ~ e C1([0, lJ; •/. Doss [21 and Sussmann [17] have shown that, under appropriate conditions on the coefficients f and g~ (i = 1 . . . . . k) [see (2) below], the mapping S which takes z=(z ~ ..... z k) into the solution x of(l) admits a continuous (w.r. to the sup-norm) extension to C([0, 1]; R~). If one wants to extend the mapping S "by continuity" to inputs, say, with jumps, an LP-norm (rather than the sup-norm) is suitable. In contrary to what is asserted in [13, Lemma 2 and Theorem 1], we show (Example 1) that under the assumptions (2) in general S does not admit a continuous extension to L~ 1]; Rk). S is, however, Lipschitz continuous with respect to LP-norm on each L~~ set (Theorem 1), hence admits a continuous extension to L~([0, 1]; R k) in this sense. Viewed as a mapping from D([0, 1]; R k) (the space of right continuous functions with left limits) into D([0, 1]; ](d), S will be shown to be continuous w. r. to the Skorokhod topology [1,7], and local Lipschitz dependence in the Skorokhod metrics will be discussed (Theorem 2). For z e D([0, 1]; R k) with finite quadratic variation, x = S(z) obeys an integral Eq. (4), which is a deterministic analogue of the stochastic integral Eq. (9) in [13] (there called "canonical extension" of (1) for semimartingale inputs). It is obvious from (9) that the mapping S gives a "pathwise solution" of a certain Stochastic differential equation, if z = z(co) is, e.g., a semimartingale. Especially, ff z