The Fundamental Theorem of Integral Calculus: a Volterra's generalization applied to flat functions
aa r X i v : . [ m a t h . HO ] A ug THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS:A VOLTERRA’S GENERALIZATION APPLIED TO FLAT FUNCTIONS
CARLO BENASSI, MICHELA ELEUTERI
Abstract.
In a recent paper [5] a smooth function f : [0 , → R with all derivatives van-ishing at 0 has been considered and a global condition, showing that f is indeed identically0, has been presented. The purpose of this note is to replace the classical FundamentalTheorem of Calculus for the Riemann integral, as it has been used in [5], with a weakerform going back to Volterra [7], which is little known. Therefore the proof we propose in thispaper turns to be important also from the teaching point of view, as long as in literaturethere are very few examples in which explicitly the lower integral and the upper integralof a function appear (usually the assumption that the function is Riemann-integrable isrequired). August 28, 2020It is well known that one of the main reasons why Lebesgue integral replaced the Riemannone in the applications is the fact that in the theory of Lebesgue integration the FundamentalTheorem of Calculus holds under weaker assumptions; in particular all absolutely continuousfunctions can be reconstructed starting from their derivative, by means of the Lebesgueintegral.In the theory of Riemann integration the most general version of the Fundamental Theoremof Calculus is the following:
Theorem 1.
Let f : [ a, b ] → R be a real valued differentiable function such that f ′ isintegrable; then, for all x ∈ [ a, b ] it holds f ( x ) = f ( a ) + Z xa f ′ ( t ) dt. When f ′ is not integrable, the Riemann integration theory allows anyway, sometimes, to getinformation on f, without having to resort to more sophisticated notion of integration, suchas for instance the Lebesgue or the Henstock one.The aim of this paper is just that of showing a situation of this kind, which we believe itmay be of interest because the Riemann integral has retained a central role from the didacticpoint of view.In particular, the main result of this paper is Theorem 3, which generalizes a result by G.Stoica [5]. In order to prove Theorem 3, we will use a very nice theorem by V. Volterra[7], see also [6] pag. 48, that is (undeservedly) little known. We notice that Theorem 1 is acorollary of Theorem 2. The authors are partially supported GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Proba-bilit`a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and by the University ofModena and Reggio Emilia.
Theorem 2.
Let h : [ a, b ] → R be a bounded function. Suppose that there exists H : [ a, b ] → R such that H ′ ( x ) = h ( x ) for all x ∈ [ a, b ] . Then Z b ∗ a h ( x ) dx ≤ H ( b ) − H ( a ) ≤ Z ∗ ba h ( x ) dx. Here we denote with R b ∗ a h ( x ) dx the lower integral and with R ∗ ba h ( x ) dx the upper integral in the Riemann formulation.A flat function f : [0 , → R is a smooth function with all derivatives vanishing at zero. Atthe beginning of the last century Denjoy and Carleman (see for instance [2] for more details)considered the class of quasi-analytic functions, being flat at 0, infinitely differentiable on[0 ,
1] and satisfying suitable inequalities involving all the derivatives, for instance | f ( n ) ( x ) | ≤ m n a n +1 , (1)for some a > m n ≥ n ∈ N and x ∈ [0 , { m n } n ∈ N for which f is identically zero. These conditions remained of afundamental importance also in the works by Bernstein and Mandelbrojt (see for instance[1], [4]) in the second half of the century.In a very recent paper [5], condition (1) has been replaced by a global simpler inequality,namely | x f ′ ( x ) | ≤ C | f ( x ) | for some C > x ∈ [0 , C − function and its firstderivative and, together with the assumption that f ( n ) (0) = 0 for every n ∈ N , can be usedto conclude that f is identically 0 in [0 , f is a C function turns tobe essential for the author in order to apply the Fundamental Theorem of Calculus for theRiemann integral.In the next theorem, which constitutes our main result, we will show that actually it is notnecessary to require that f ′ is a Riemann-integrable function. It can be stated as follows: Theorem 3.
Let f : [0 , → R be a real valued differentiable function such that f (0) = 0 (2) and ∀ n > ∃ δ n > | f ( x ) | < x n ∀ x ∈ ]0 , δ n [ . (3) Assume moreover that ∃ C > | xf ′ ( x ) | ≤ C | f ( x ) | ∀ x ∈ [0 , . (4) Then f ( x ) = 0 for every x ∈ [0 , . In order to prove this result we will apply Theorem 2 with h = | f ′ ( x ) | . We will provetherefore that (2)–(4) entails that h is bounded so we are not assuming any integrability onour function h . Condition (2) actually turns out to be equivalent to f ( n ) (0) = 0 if all thesederivatives exist. VOLTERRA’S GENERALIZATION APPLIED TO FLAT FUNCTIONS 3
Remark . As we already mentioned before, what we believe it is interesting to remarkis that in the proof of Theorem 3 only the Riemann integral calculus has been employed.However, the proof could have been simplified if we would have used a more powerful tool.For instance, by means of the Henstock-Kurzweill integral [3], Volterra’s theorem would havebeen replaced by the Fundamental Theorem of Calculus (if we interpret the integrals in thesense of Henstock-Kurzweil, given a real valued differentiable function h , one always has that Z ba h ′ ( s ) ds = h ( b ) − h ( a ) ! . Proof.
First of all we observe that | f ′ | is bounded. Indeed, condition (2) and (4) imply that f ( x ) x is bounded (it is a continuous function which has limit in 0) therefore also f ′ is alsobounded.Let us set g ( x ) := | f ( x ) | Obviously g is differentiable in all points x for which f ( x ) = 0. On the other hand, if x is suchthat f ( x ) = 0, then also g turns to be differentiable because in these points f ′ ( x ) = 0 (thisfact can be deduced from (3), if x = 0 and from (4) if x ∈ ]0 , g is differentiablein the whole interval [0 ,
1] and in particular we have | g ′ ( x ) | = | f ′ ( x ) | ∀ x ∈ [0 , | g ′ ( x ) | is bounded.At this point, by Theorem 2 we show that f ≡ , ∀ n > C, | f ( x ) | < x n ∀ x ∈ ]0 , . Notice that condition (3) holds only in a small interval ]0 , δ n [. Notice that the constant C isthe one appearing in (4).Suppose by contradiction that this does not hold, i.e. suppose that there exists n > C suchthat f ( x ) ≥ x n for all x ∈ ]0 , n > C such that the set E n := { x > | f ( x ) | = x n } is non-empty. From (3) we deduce that E n ⊂ [ δ n , E n is closed as long as f is acontinuous function, therefore it has a minimum point¯ x n := min E n > . Then, in ]0 , x n [ , we have | f ( x ) | < x n and so, by Theorem 2 applied to the function g (¯ x n ) n = | f (¯ x n ) | = g (¯ x n ) − g (0) ≤ Z ∗ ¯ x n g ′ ( x ) dx ≤ Z ∗ ¯ x n | g ′ ( x ) | dx = Z ∗ ¯ x n | f ′ ( x ) | dx ≤ Z ¯ x n C | f ( x ) | x dx < C Z ¯ x n x n x dx = Cn (¯ x n ) n < (¯ x n ) n . This tells us that E n = ∅ i.e. ∀ n > C, | f ( x ) | < x n ∀ x ∈ ]0 , . This entails that | f ( x ) | ≡ . (cid:3) C. BENASSI, M. ELEUTERI
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