In addition to uniqueness, another consequence of the inequality in (45) is the fact that solutions of the differential equation
depend continuously on the initial value x(a); that is, if and are two solutions of (47) on the interval such that the initial values and are sufficiently close to one another, then the values of and remain close to one another. In particular, if then (45) implies that
for all t with Obviously, we can make as small as we wish by choosing 1 sufficiently close to zero.
This continuity of solutions of (47) with respect to initial values is important in practical ...