By N. Bourbaki, P. Spain
This is an English translation of Bourbaki’s Fonctions d'une Variable Réelle. assurance contains: features allowed to take values in topological vector areas, asymptotic expansions are handled on a filtered set outfitted with a comparability scale, theorems at the dependence on parameters of differential equations are at once appropriate to the examine of flows of vector fields on differential manifolds, etc.
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I] (same method. using I. p. 22. corollary). 7) Let f be a finite real function 11 times differentiable at the point Xo. and g a vector function which is 11 times differentiable at the point Vo = f(xo). Let f(xo g(yo + h) + k) = lIo + Qlh + ... + (I"h" + r,,(h) + ... ) = b o + b,k be the Taylor expansions of order /I of J and g at the points Xo and Yo respectively. Show that the sum of the 11 + I tenns of the Taylor expansion of order Il of the composItc function go f at the point Xu is equal to the sum of the terms of degree ~ II in the polynomial bo + b,(a,h + ...
Y) runs through the set of pairs of distinct points of 1 (cf. exerc. 4a». 6) On the interval 1 = [- I. +1] consider the vector function f, with values in R", defined as follows: f(t) = (0,0) for -I :( t :( 0; fU) = ( , I r, 1) 1- sin" cos, for 0 :( t :( I. Show that f b differentiable on ] - I, + I [ but that the image of thb mterval under f' IS not a connected set in R2 (et exerc. 4 c)). 7) Let f be a continuous vector function defincd on an open interval I C R. with values in a normcd space E over R.
4) Let I be a finite real function, continuow, on a compact interval [II, h] in R. and baving a right derivative at every point or the open interval ]a. /J[. Let 111 and M be the greatest lower bound and least upper bound (finite or not) of over ]a, b[. t;; a) Show that when x and y run thlllUgh ]11, /J[ keeping x of. 1') contains ]111. M[ and is contained in [111. M]. ite sign at the two point, c, d of la. I takes the same value). h) If. rurther, of l has a left derivative at every point of ]a.