A Variational Definition for Limit and Derivative

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1 Amerian Journal of Applied Mathematis 2016; 4(3): doi: /j.ajam ISSN: (Print); ISSN: X (Online) A Variational Definition for Limit and Derivative Munhoz Antonio Sergio 1, *, Souza Filho 2, Antonio Calixto 1, 2 1 Center of Mathematis, Computation and Cognition, Federal University of ABC, Santo André, Brasil 2 Shool of Arts, Sienes and umanity, University of São Paulo, São Paulo, Brasil address: antoniosergiomunhoz@gmail.om (M. A. Sergio), asouzafilho@usp.br (S. Filho) * Corresponding author To ite this artile: Munhoz Antonio Sergio, Souza Filho, Antonio Calixto. A Variational Definition for Limit and Derivative. Amerian Journal of Applied Mathematis. Vol. 4, No. 3, 2016, pp doi: /j.ajam Reeived: Marh 12, 2016; Aepted: May 11, 2016; Published: May 22, 2016 Abstrat: Using the topologial notion of ompaity, we present a variational definition for the onepts of limit and derivative of a funtion. The main result of these new definition is that they produe implementable tests to hek whether a value is the limit or the derivative of a differeniable funtion. Keywords: Variational, Limit, Derivative, Differeniation 1. Introdution Limit is a fundamental onept in mathematis whih priniples date bak by the time of the method of exhaustion invented by Eudoxo ( ac). The strengthening of the onept starting from Newtonian Physis at XVIII entury, and a preise definition was originated as a result of the ontributions of mathematiians like Cauhy, Bolzano and Weirstrass of the XIX entury (see [2] for disussion about this subjet and original referenes). The started definition was formerly presented by Weirstrass and has been known as the ε and δ definition, see [5]. Apparently, it has never been hanged essentially and seems to be the only one available. The first ontribution of this paper is to bring an alternative definition for limit whih, roughly speaking, an be resumed as the best loal approximation for a funtion. We do this in theorem 1 where we show that the value L lim is the best loal approximation suh that, for any value V, V L, f(x)-v > f(x)-l, for any x lose to p. Also, without using the usual haraterization of limit, but rather the so alled best linear loal approximation for a funtion, we propose to obtain an alternative definition for the derivative of a funtion. By Theorems 2 and 3, we see that a differentiable funtion at p with domain and image in a finite dimensional linear spae is the one whose quotient is loally bounded at p, and with a linear operator L as the best loal approximation, suh that: f(x)-f(p)-l(x-p) f(x)-f(p)-t(x-p), for all linear operators T and also Lf (p). The insight for Theorems 2 and 3 was given as follows: take a urve and the tangent line in a point of it. Any other line is a worse loal approximations for the urve. For example, for f(x)x 2 and p0 we an get that: x 2 x 2 -ax, for x < a /2. Finally, in the last setion we suggest a possible gain in using the presented definition of derivative to obtain a numerial derivative approximation. In the next setion, we work in a general setting.

2 Amerian Journal of Applied Mathematis 2016; 4(3): Definition of Limit Let X and Y be metri spaes with distanes denoted by and B(r,s) the ball with enter at s and radius r. The first theorem is an alternative definition of limit. Observe that the minimum ondition on the funtion is its image in a proper metri spae, that is, a spae whih losed bounded sets are ompat. The main innovation is the variational formulation. Theorem 1. Let X be a metri spae and Y a proper metri spae. Denote by f:x Y a funtion and p an aumulation point of X. The value L is the limit if and only if f is loally bounded at p and f(x)-l < f(x)-v, (1) for any V Y, V L, and x n(p), where n(p) is a neighborhood of p, depending on the hoie of V. Proof. If f is loally bounded, we an take a ball B X with enter at p and another ball B Y with enter at 0 suh that f(x) B' if x B. Suppose, by the way of ontradition, that does not exist or that ". In any ase, there are # >0 and a sequene (x n ) X, n 1, suh that x n p, if n + and f(x n )-L >#, Without loss of generality we assume x_n B. From (1), for all V Y, we an take n V suh that f(x n )-V > # when n>n V, where n V depends on V. As B' is a bounded set and Y is a proper metri spae, there is a finite list of balls B(V i, # ) 1 i N, whih overs B': From above, take n i so that +, - +. /,# 12/23 (2) f(x n )-V i >#, when n>n i. Set /,1 / 38. If n>4, we have: f(x_n)-. / ># for any i. Therefore, f( 4 ) - 12/23 +. /,# if n>n **, whih ontradits (2). The onverse is standard. 3. Definition of Derivative In this setion, we state an alternative definition for the onept of derivative. Let X, Y be normed linear spaes with norm denoted by, U X an open set and f:u Y, a funtion. The following disussion moves our investigation. It is well known that if X is the real spae, the derivative an be defined as f p. So, by Theorem 1, fx fp f px p fx fp Vx p, for any V and x n(p), where the neighborhood depends on the hoie of V. Therefore, the derivative is the best linear approximation for a funtion. A generalization of this interpretation is not so easy. We shall obtain it in two steps. First, in the next theorem, where the spae of domain has finite dimension and additionally the funtion is ontinuous in a neighborhood of the point. Seond, in the last theorem, the onverse is obtained, but both spaes of the domain and image have finite dimension. Theorem 2. Let X, Y be normed spaes, U X, an open set and suppose X is of finite dimension. Suppose also that the funtion f:u Y is ontinuous in a neighborhood of p U. So f is differentiable at p and has derivative f'(x)l only if (f(x)-f(p))/ x-p is loally bounded at p and fx fp Lx p fx fp Tx p, (3) for all T L(X,Y) and x n(p), where the neighborhood n(p) depends on the hoie of T. Proof. Let f be differentiable with derivative f'(p)l. The first part of the theorem is obvious. Assume, by the way of ontradition, that (3) is not true. For some A, we an get a sequene ( 4 ) U, n 1, 4 p, if n +, suh that: Set and f( 4 )-f(p)-l( 4 -p) > f( 4 )-f(p)- A ( 4 -p), (4) G(x)f(x)-f(p)-L(x-p), (x)f(x)-f(p)- A (x-p) F(x) G(x) - (x). From the ontinuity of x G(x) and x (x), the funtion x F(x) is also ontinuous. As F( 4 )>0, n 1, and ( 4 ) is a ompat set, we have F( 4 )>r, n 1, for some r>0. Sine the unit ball of X is ompat, for any n there exists B 4 suh that B 4 -p 4 -p and C A"DB 4EF C A " (5) B 4 E We laim that (4) is true if we replae 4 by B 4, for a larger n, if neessary. As 4 p and B 4 p if n +, we an assert that G( 4 )-G(B 4 ) < G,

3 139 Munhoz Antonio Sergio et al.: A Variational Definition for Limit and Derivative if n>n., for some n.. Similarly, ( 4 )-(B 4 ) ) < G, if n>n.., for some n.. and there is no loss of generality in assuming n.. n.. We already know that: G( 4 ) - ( 4 ) >r, Combining the inequalities, we have G(B 4 ) > (B 4 ), (6) if n>n... Finally, take s (0, T * -L ). From the differentiability, we an find n..., n... n.., suh that: C B 4 B 4 if n>n.... By (6) and (7), both L B 4 and T * B 4 B( B 4 / B 4, J ) if n>n... Therefore: " B 4 B 4 C I J B 4 B 4 are in the ball L B 4 " B 4 B 4 B 4 LIJ A ontradition of (5). Theorem 3. Let X, Y be normed linear spaes of finite dimension and UX, an open set. The funtion f:x Y is differentiable at p and f'(p)l if (f(x)-f(p))/ x-p is loally bounded at p and fxfplxp ) fxfptxp, for any TM(X,Y) and xn(p), where the neighborhood n(p) depends on the hoie of T. Proof. Suppose, ontrary to our laim, that f is not differentiable at p or f'(p) L. Consequently, there are # $ N 0 and a sequene ( 4 )U, n 1, 4 p, if n +, suh that: (7) P Q 4 QER 4E SN# $ (8) Sine x (f(x)-f(p))-l x-p, then (f(x)-f(p))/ x-p is loally bounded at p, without loss of generality we an assume that Q 4 QE 4 E )R, n 1for some R>0. From (8) it follows immediately that, for any TM(X,Y), C 4 A 4 CN# $ (9) if n>n T, where n T depends on the hoie of T. Next, we onstrut a list of operators whih do not satisfy (9) obtaining a ontradition. Obviously, -VUW,X UV,# $ overs B0,R Y. As Y is loally ompat, there are N>0 and y j, 1) j)n, suh that: Denote S{ xx, x 1}. For any xs and B Z, 1<j<N, take A,B Z M[,\ suh that A,B Z B Z. Sine ]^A,B Z] is ontinuous, we an find _,B Z N0 with the property that A,B ZD+,_,B ZF+B Z,# $ Let _ be the smallest of _,B Z,1 ) Z )3. Obviously, `a+,_. ` From the ompaity of S, there are M>0 and a list 1,,, suh that ` a + /,_ /. 12/2 Set A /,B Z as A /Z. By (9), there exists 4 /Z suh that C 4 A /Z 4 C N# $, if 4N4 /Z. Setting /Z,1 ) /),1 )Z)3, 1 we have C 4 A /Z 4 C N# $ (10) if 4N4., for all i,j. For any n 1 and suitable i,j, we have +D /,_ / F and ene, 4 L 4 4 whih ontradits (10). UV d,e $ 4. Numerial Derivative A /Z 4 LI# 4 $, 4 The definition of derivative of a funtion as the best linear approximation provides a method to obtain its numeri value. Considering the question more losely, we see that differentiability for real funtions is not enough and more regularity is neessary. In spite of it, there is some gain. In fat, we show here that it is possible, by that definition, to get a better approximation for numeri derivative than the usual methods. In general, numerial methods for derivative alulation

4 Amerian Journal of Applied Mathematis 2016; 4(3): have error estimations whih depend on the mahine preision. As an example, onsider the numerial derivative of a real funtion alulated by the finite differene method: +g g g Aording to [3], hapter 5, if f is of lass h i, this i approximation has an error of order # when the hoie of h is optimal, that is, when h is of order # jj 1 i. For a funtion f, # is of the same order as # k where it is the mahine preision. But in general # is greater than # k. ene, we an hoose a subset B for the last theorem. Nevertheless, relative to h and T where h r and T B, the numerial minimization of an expression remains impossible to be done omputationally. We remove this diffiulty, going to another method. Theorem 4. Let X, Y be linear finite dimensional spaes, U X an open set and f:u Y of lass h. Take p U and all f'(p)m. Setting k 1 and k two approximations of m suh k k 1 k k. If f(p+h)-f(p)-k 1 h f(p+h)-f(p)- k h, (11) for any h, g G, requiring only that r satisfies G < 1 n k k 1 k k n, where ko p,, (]) for ] G. Then k k 1 < k k. Proof. Observe that the spae of the norm an be obtained by the ontext. For instane, (k k 1 )g (k k 1 )g \, g g [, (k k ) (k k ) M([,\),,, (]),, (]) M([,\), (see [1]). From (11): f(p+h)-f(p)-mh+(m-k 1 )h < f(p+h)-f(p) -mh+(m-k )h. But f is h so: (+g) () kg where ] is between p and p+h. So,, (])g,, (])g,, (])g L +(k k 1 )gl < L +(k k )gl, and hene q(k k 1 ) g g q<q(k k ) g g q+,, (]) g. This gives that: q(k k 1 ) g g q< (k k ) +,, (]) g. Sine X has finite dimension, there is g 7g g G} suh that (k k 1 ) q(k k 1 ) g g q And the last two relations give: (k k 1 ) < (k k ) +,, (]) g. This fores that k k 1 > (k k ). In fat, if the opposite was true, we would have: and so,, (]) g >n (k k 1 ) (k k ) n g > n (k k 1) (k k ) n that is ontrary to the assumption that g G. The last proposition gives a test to deide, between k 1 and k, whih of them is loser to the derivative m omparing the residues. But the riterion g < n (k k 1) (k k ) n is not pratial beause uses the value of the unknown m. Nevertheless, using the test when the riterion is false will not produe worse results. In this ase, n (k k ) (k k 1 ) n g So, if M is not so big, the differene between the errors k k 1 and k k is of order h. We now onsider an example to show how using this test. Set f(x)x^3. By the finite differene sheme, supposing # k of order 1W r and using h of order 1W r i, the error will be of order 1W 1s i. To alulated the residues, we observe that f(1) , f(1+h) The tables 1 and 2 show firstly that the best approximation among , ,, is Next, the seond table shows that the best approximation among ,, is So the approximation has 7 orret digits. To larify the proedure used in the last paragraph, it may be onvenient to do the following observation. If k 1 is a better approximation for m than k (or k 1 is loser than k to m) then k k 1tk when k 1 >k. and k k 1tk when k 1 <k This gives the algorithm whih we shortly desribe below. For simpliity, suppose that m0.d 1 00*10 f and it is known that the digit d 1 is orret. Our aim is to obtain the seond and the third orret digits in the following steps. 1. Obtain the best approximation among the values 0.d 1 t0*10 f for t0,1,,9 and set it as d.

5 141 Munhoz Antonio Sergio et al.: A Variational Definition for Limit and Derivative 2. Consider the ase options: Case d0. Obtain the best linear approximation among 0.d_10t*10 f for t0, 1, 2, 3, 4, 5; set it as d 3 and do d 2 0. Case 1 v w. Obtain the best linear approximation among the data 0.d_1(d-1)t 1 *10 f for t 1 5, 6, 7, 8, 9 and among the data 0.d 1 dt 2 *10 f for t 2 0, 1, 2, 3, 4, 5. If the best approximation of the eah data group are in the first one, then d 2 d-1 and $d 3 t 1. Otherwise d 2 d and d 3 t 2. Case d9. Obtain the best linear approximation among data $0.d 1 8t*10 f for t5, 6, 7, 8, 9 and $0.d 1 90*10 f. If the best approximation is in the first group, set d 2 8 and d 3 t. Otherwise, set d 2 9$ and d 3 0. In general, alling as the hyperplane equidistant from k 1 and k, so m is in the side of that ontains k 1. The tables forward shows the numerial results. m Table 1. Residue alulation. f(1+h)-f(1)-mh * * * * * * *10-7 m Table 2. Residue alulation. f(1+h)-f(1)-mh * * * * * * Conlusion We dealt with the definition limit and derivative of a funtion, whih is well known, it is a historially intriate question. Thus, we proposed alternatives definition of these onepts that have, as mainly differenes of the standard ones, the fat they are implementable forms and also that suh definitions drop out the ε of the δ-ε usual definitions. Referenes [1] Dieudonne, J., Foundations of Modern Analysis, Pure and Applied Mathematis \textbf{10}, Aademi Press, New York, [2] J. V. Grabiner, J. V., Who Gave You the Epsilon? Cauhy and the Origins of Rigorous Calulus, The Amerian Mathematia Monthly (90) (3), , [3] W.. Press, W.., et al, Numerial Reipes: The Art of Sientifi Computing, Cambridge University Press, New York, [4] F. Riesz, F., Oeuvres omplètes, 2 vol, Paris, Gauthier-Villars, [5] J. V. Grabiner, J. V., The Changing Conept of Change: the derivative from Fermat to Weistrass, Mathematis magazine, vol 56, no. 14, pp , [6] Rudin, Walter., Funtional Analysis, M Graw-ill, NY, [7] Boyer, C. B., The istory of Calulus and Its Coneptual Development, Dover, N Y, [8] enry, D. B., Differential Calulus in Banah Spaes, Dan enry s Manusript, http// [9] Kato, T., Pertubation Theory for Linear Operators, Springer-Verlag, N Y, [10] Bordr, K. C. and Aliprantes C. D., Infinite Dimension Real Analysis, Springer-Verlag, N Y, 2006.