By Swokowski

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70) then ipo is of class in x, and thus it is of class C^. To that effect, consider the functions ipi and given by (pi{t,x) = x and $i(t,a:) = Id. 71) — {s, cpn{s , x)) ^n{s , x) ds . 73) = ld + ^{s,(pn{s,x))^^{s,x)ds. 3. 73). This implies that (pn+i is of class C^. 74) for each n € N. Clearly, d(pi (t,x) = Id = $ i(t, x). 74) holds for a given n. 72) that d(Pn+l , f df (t,a;) = Id-f- J ■^{s,(pn{s,x))^n{s,x)ds = ^n+l{t>x). dx This yields the desired identity. Finally, given t G {to —a,to + oc), we consider the sequence of functions fn{x) = fnitt x).

85). 52. 85), then { xq} is an orbit of this equation. P ro o f. 85). 53. 45, the equation x' = x^ has the solutions x{t) = 0, with maximal interval E, and x{t) = l / ( c — t), with maximal interval (—oo, c) or (c, +oo), for each c € E. Thus, we obtain the orbits {0 } (coming from the critical point 0), { l / ( c —t ) : t G (—o o ,c )} = E'*’ and { l / ( c — t) : t G { c , + o o )} = E “ . Now we show that in a sufficiently small neighborhood of a noncritical point one can always introduce coordinates with respect to which the orbits are parallel line segments traversed with constant speed.

We first establish an auxiliary result on the limit of a sequence of differ entiable functions. 41. If a sequence of differentiable functions fn- U — R " in an open set 17 C R” is (pointwise) convergent and the sequence dfn of their derivatives is uniformly convergent, then the limit of the sequence fn is differentiable in U and its derivative is the limit of the sequence dfn in U. P roof. Let f : U —^W^ and g \ U - ^ Mn be, respectively, the (pointwise) limits of the sequences / „ and dfn, where Mn is the set of n x n matrices 24 1.