By Nicolas Bourbaki, Philip Spain (auth.)

This publication is an English translation of the final French version of Bourbaki’s *Fonctions d'une Variable Réelle.*

The first bankruptcy is dedicated to derivatives, Taylor expansions, the finite increments theorem, convex services. within the moment bankruptcy, primitives and integrals (on arbitrary periods) are studied, in addition to their dependence with appreciate to parameters. Classical services (exponential, logarithmic, round and inverse round) are investigated within the 3rd bankruptcy. The fourth bankruptcy provides an intensive therapy of differential equations (existence and unicity houses of recommendations, approximate options, dependence on parameters) and of platforms of linear differential equations. The neighborhood examine of capabilities (comparison family members, asymptotic expansions) is handled in bankruptcy V, with an appendix on Hardy fields. the idea of generalized Taylor expansions and the Euler-MacLaurin formulation are awarded within the 6th bankruptcy, and utilized within the final one to the learn of the Gamma functionality at the genuine line in addition to at the advanced plane.

Although the themes of the booklet are regularly of a sophisticated undergraduate point, they're awarded within the generality wanted for extra complicated reasons: capabilities 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 acceptable to the research of flows of vector fields on differential manifolds, etc.

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Show that there exists a point x ∈ ]a, b[[ ∩ A such that f(b) − f(a) fd (x) (b − a). h, and use th. 2 of I, p. ) § 3. 1) With the same hypotheses as in prop. 2 of I, p. g( p) ]. p 2) With the notation of prop. 2 of I, p. y] 0 for all y ∈ F implies that a 0 in E. Under these conditions, if gi (0 i n) are n + 1 vector 40 Ch. f (n) ] 0 then the functions gi are identically zero. 3) With the notation of exerc. f (n) ], n) without ambiguity (exerc. f (n) ] identically. 4) Let f be a vector function which is n times differentiable on an interval I ⊂ R.

Let f i (1 and ci (1 i i p) be p convex functions on an interval I ⊂ R, p p) be p arbitrary positive numbers; then the function f ci f i i 1 is convex on I. Further, if for at least one index j the function f j is strictly convex on I, and c j > 0, then f is strictly convex on I. This follows immediately by applying the inequality (1) (resp. (2)) to each of the f i , multiplying the inequality for f i by ci , and then adding term-by-term. PROPOSITION 3. Let ( f α ) be a family of convex functions on an interval I ⊂ R; if the upper envelope g of this family is ﬁnite at every point of I then g is convex on I.

PROPOSITION 1. Let f be a ﬁnite real function, convex (resp. strictly convex) on 2 distinct points of I , and an interval I ⊂ R. For every family (xi )1 i p of p every family (λi )1 i p p of p real numbers such that 0 < λi < 1 and λi 1, we i 1 have p (resp. p λi xi f i 1 i 1 p p λi xi f < i 1 λi f (xi ) (3) λi f (xi )). (4) i 1 Since the proposition (for convex functions) reduces to the inequality (1) for 2 we argue by induction for p > 2. The number μ p p−1 i 1 λi is > 0; it is immedip−1 ate that if a and b are the smallest and largest of the xi then a in other words, the point x implies that μf (x) p−1 1 μ p−1 p−1 λi xi i 1 λi b; i 1 λi xi belongs to I, and the induction hypothesis i 1 λi f (xi ); moreover we have, from (1), that i 1 p p λi xi f f (μx + (1 − μ)x p ) μf (x) + (1 − μ) f (x p ) i 1 λi f (xi ).