By N. Bourbaki, P. Spain

This is an English translation of Bourbaki’s Fonctions d'une Variable Réelle. insurance contains: capabilities allowed to take values in topological vector areas, asymptotic expansions are taken care of on a filtered set built with a comparability scale, theorems at the dependence on parameters of differential equations are without delay acceptable to the examine of flows of vector fields on differential manifolds, etc.

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1 +, r) --. (O'X + f:») = O. SUdl that § 4. \". Y In I. Show that If j is bounded above on one open interval ]a. b[ contained in I. then f is convex on I (show first that f is bounded above on every compact Interval contained in l. then that f is contInUOUS at e\'cry interior point of I). I I) Let I be a contmuous functIon on an open Interval I C R. having a tlmk nght del ivative at every point of I. I'. \". I(x». (yl-f(X)j' ) .! \ < y . v-x Give an example of a function which is not convex. has a tinite right denvative everywhere.

The last term rn (x) = u(x )(x - a)" In! is called the remainder in the Taylor formula of order 11. lorder II + 1 on I. one can estimate II r ll (x) II in temlS of thiS II + 1ril derivative, on all of I, and not just on an unspecified neighbourhood of a: PROPOSITION 3. flllfln+')(x)11 ~ M on 1, thell we have IIrll('y)1I ~ Ix _ al"+ 1 (9) M --(II + I)! on 1. on Indeed, the formula holds for 11 = O. by I, p. 15. tho 2. Let us prove it by induction by the induction hypothesis applied to f', one has II : Iv Ilr;,(y)11 ~ M .

Let f be a convex (resp. x)jimcfion on I; bare nl'O interior points of I such that (/ < b one has (fig. 3) a b () if a and :r Fig. 3 ,'( ):<: fib) - f(a) :<: t"(I]) " " , !! b-a Jd (/ (8) (resp. ,'() Jd a < f(h) - f(a) < "(b». J, b-a /I' (9) §4. CONVEX FUNCTIONS OF A REAL VARIABLE 29 The double inequality (8) results from (6) and (7) by a simple change of notation. On the other hand, if f is strictly convex and c is such that a < e < b one has, from (8) and prop. 5, f~(a) ~ fie) - f(a) fib) - feu) < feb) - f(c) ~ f'(b) ----- < b -c I?