C Dependence on Initial Conditions for ODE

Lemma 0.1 Let (X, d1) and (Y, d2) be metric spaces. Form the metric space (X × Y, d) where d ((x1, y1), (x2, y2)) = d1(x1, x2) + d2(y1, y2). Suppose that for each x ∈ X we have a function ux ∈ Cb(Y ), the space of bounded continuous functions from Y to R . (Here l is fixed.) Suppose moreover that for each x0 ∈ X, lim x→x0 ‖ux − ux0‖ = 0. (Here ‖‖ denotes sup norm on Y .) For x ∈ X, y ∈ Y , put u(x, y) = ux(y). Then h : X×Y → R l