Bulletin of TICMI Vol. 8, No., 204, 93 09 On the Differentibility of Rel, Complex nd Quternion Funtions Omr Dzgnidze A. Rzmdze Mthemtil Institute of Iv. Jvkhishvili Tbilisi Stte University, 6 Tmrshvili St., 077, Tbilisi, Georgi (Reeived Deember 26, 203; Revised Mrh 25, 204; Aepted June 6, 204) In the pper, the neessry nd suffiient onditions re estblished for the ontinuity of funtions of severl vribles, for the differentibility of funtions of severl rel vribles, nd for the C n -differentibility of funtions of severl omplex vribles. Also, the rules re given for lulting H-derivtives of quternion funtions, nd the neessry nd suffiient onditions re obtined for the H-differentibility of quternion funtions. Keywords: Seprtely strong (ngulr) prtil ontinuity, nonintense ngulr prtil ontinuity, ngulr (strong) prtil derivtive, ngulr (strong) grdient, nonintense ngulr prtil derivtive, differentibility, totl differentil, vrition in the Hrdy (Arzel) sense, indefinite integrl, bsolutely ontinuity, Lebesgue s intense point, definite integrl with prmeter, C n -differentibility, Hrtog s min theorem, H-derivtive AMS Subjet Clssifition: 32A20, 30G35, 30A05, 30B0. Introdution A funtion of mny vribles will not hve the ontinuity or differentibility property only beuse it hs the sme property with respet to eh independent vrible. Funtions with this drwbk t individul points hve been known sine the lte 9th entury, nd on the mssive set sine the 20th entury. Nmely, the following sttement is vlid. Sttement A ([, pp. 432-433]): There exists the funtion of two vribles whih is disontinuous t lmost every point of the unit squre nd t every point of tht squre ontinuous with respet to every vrible. Note tht this Tolstov s funtion does not posses lmost everywhere even the property of ontinuity on the whole (see [2]; [3, pp. 32 39]). These nd nlogous problems re studied e.g. in Z. Piotrowski [4]. Here the problem onsists in finding out whether there exists or does not exist ny notion of funtion with respet to n independent vrible nd whether the fulfillment of this notion for ll independent vribles will be the neessry nd suffiient ondition for the ontinuity nd differentibility of the funtion itself. In this pper, the disussion onentrtes on this problem. Emil: odzgni@rmi.ge ISSN: 52-0082 print 204 Tbilisi University Press
94 Bulletin of TICMI In formulting the min results, we use the following nottion: x = (x,..., x n ), = (,..., x0 n), x( k ) = (x,..., x k, k, x k+,..., x n ).. The onditions for the ontinuity. The funtion f is lled strong prtil ontinuous with respet to the vrible x k t the point, if the equlity [f(x) f(x( k ))] = 0 (.) x is fulfilled nd f is lled seprtely strong prtil ontinuous t the point, if f with respet to every vrible is strongly prtil ontinuous t, i.e. equlity (.) is fulfilled for ll k =, 2,..., n. Theorem. ([5]; [2]; [3, pp. 20 25]): For the ontinuity of the funtion f t the point, it is neessry nd suffiient tht it possesses seprtely strong prtil ontinuity t. 2. The expression f(x) f(x( k )) for x j j j x k k, j k, depending on the vribles x,..., x n, is lled n ngulr prtil inrement of the funtion f t the point with respet to the vrible x k, orresponding to the olletion = (,..., k, k+,..., n ) of positive onstnts. The ngulr prtil ontinuity of the funtion of t the point with respet to the vrible x k mens the fulfillment of the equlity x k k [f(x) f(x( k ))] = 0 (.2) x j j j x k k j k for every olletion = (,..., k, k+,..., n ) of positive onstnts. The funtion f is lled seprtely ngulr prtil ontinuous t the point, if with respet to every vrible the funtion f possesses the property of ngulr prtil ontinuity t the point, i.e. if for ll k =,..., n nd for every olletion = (,..., k, k+,..., n ) of positive onstnts, equlity (.2) holds. Theorem.2 ([5]; [2] ; [3, pp. 25 27]): For the ontinuity of the funtion f t the point, the neessry nd suffiient ondition is the seprtely ngulr prtil ontinuity t. 3. If in the definition of the ngulr prtil ontinuity we put j = for ll j k, then we hve the nonintense ngulr prtil ontinuity t the point of the funtion f with respet to the vrible x k. Theorem.3 ([3, pp. 27 28]) : For the ontinuity of the funtion f t the point, the neessry nd suffiient ondition is the seprtely nonintense ngulr prtil ontinuity of the funtion f t the point.
2. Angulr prtil derivtive nd ngulr grdient Vol. 8, No., 204 95 The existene of finite prtil derivtives of ll orders, i.e. ordinry grdients of the rel funtion f t the point does not imply the differentibility of the funtion f t the sme point. Even the funtion, possessing finite grdient t the point, my be disontinuous t. Suh, for exmple, re t the point (0, 0) the most of the funtions of two vribles indited in Piotrowski s work [4]. It is remrkble tht this ft n be relized t ll points of set, whose plne mesure is rbitrrily nerly to totl mesure. Sttement B ([, 4]): For every number µ < there exists the funtion F, defined on the squre Q = {(x, y) R 2 ; 0 x 2, 0 y }, possessing t ll points of the Q finite prtil derivtives of ll orders, nd t the sme time F is disontinuous on ertin set E Q of plne mesure µ 2. We sy tht the funtion F hs t the point n ngulr prtil derivtive with respet to the vrible x k, symbolilly f x k ( ), if for every olletion = (,..., k, k+,..., n ) of positive n onstnts there exists n independent of the finite it f x k ( ) = f(x) f(x( k )) x k k x k. (2.) x j j j x k k k j k The existene of f x k ( ) implies existene of the prtil derivtive f x k ( ), nd the equlity f x k ( ) = f x k ( ). To show this, we hve to put in (2.) x j = j for ll j k. The existene of the ngulr prtil derivtive does not, in generl, follows from existene of the ordinry prtil derivtive. If f x k ( ) is finite, then the funtion f with respet to the vrible x k hs the property of ngulr prtil ontinuity t the point. If there exist finite f x k ( ), k =,..., n, then we ll f the funtion possessing n ngulr grdient t the point nd write nggrd f( ) = (f x ( ),..., f x n ( )). Theorem 2. ([5]; [6]; [3, pp. 60 64]): For the funtion f to be differentible t the point, it is neessry nd suffiient tht nggrd f( ) is finite. The totl differentil df( ) of the differentible t the point funtion f dmits the following representtion n df( ) = f x k ( ) dx k. k= Theorem 2.2 ([6]; [3, p. 65]): For the funtion f to be differentible t the point, it is neessry nd suffiient tht the nonintense ngulr prtil deriv-
96 Bulletin of TICMI tives D xk f( ) = re finite for ll k =,..., n. f(x) f(x( k )) x k k x k x j j x k k k j k Corollry 2.3 ([6]; [3, p. 65]) : of ll f x k ( ), nd the equlity The finiteness of ll D xk f( ) implies finiteness f x k ( ) = D xk f( ), k =,..., n, n df( ) = D xk f( ) dx k. k= 3. Exmples on the differentibility Using Theorem 2.2, we n estblish the differentibility s well s nondifferentibility of onrete funtions. On the differentibility we investigte some ppering frequently funtions. Proposition 3. ([6]; [3, p. 66]): Suppose the numbers α j re positive, j =,..., n. Then the ondition α + α 2 + + α n > is neessry nd suffiient for the everywhere ontinuous funtion φ(x,..., x n ) = x α x 2 α2 x n α n to be differentible t the point = (0,..., 0). In prtiulr, the funtion γ(x,..., x n ) = ( x x n ) α is differentible t the point if nd only if α > /n. If α + α 2 + + α n, then ll D xk φ( ) re devoid of existene. Proposition 3.2 ([3, p. 67]): the funtion Suppose the numbers β j >, j =,..., n. Then n x j βj for ll rtionl x j Φ(x,..., x n ) = j= 0 t the remining points (3.) is differentible t the point = (0,..., 0), dφ( ) = 0 nd disontinuous t ll the remining points (x,..., x n ) (0,..., 0).
Proposition 3.3 ([3, p. 68]): Vol. 8, No., 204 97 ( n q x j ) for ll rtionl x j Ψ(x,..., x n ) = j= 0 t the remining points possesses the sme properties s the funtion (3.). Proposition 3.4 ([3, p. 68]): (3.2) The orresponding to the number α > 0 funtion ( n ) +α x 2 2 j for ll rtionl x j ω(x,..., x n ) = j= 0 t the remining points possesses ll properties of funtions (3.) nd (3.2). Proposition 3.5 ([3, p. 69]): g(x, x 2 ) = The funtion { x x 2 sin x x 2 for x x 2 0 0 for x x 2 = 0 is differentible t the point = (0, 0), nd its grdient grd g(x, x 2 ) is indeterminte in the puntured neighborhood of the point. Proposition 3.6 ([3, pp. 69 70]) : ψ(x, x 2 ) = possesses the following properties: The orresponding to the number q > funtion { x 2 x 2 The funtion x 2 +x 2 2 for x 2 + x2 2 > 0, 0 for x = 0 = x 2 ) ψ(x, x 2 ) is ontinuous everywhere; 2) grd ψ(x, x 2 ) is finite everywhere; 3) ψ(x, x 2 ) is not differentible t the point (0, 0); 4) grd ψ(x, x 2 ) is not ontinuous t the point (0, 0). 4. A strong prtil derivtive nd strong grdient We sy tht the funtion f possesses t the point strong prtil derivtive with respet to the vrible x k, symbolilly f [x k ] (x0 ), if there exists finite it f [x f(x) f(x( k ] (x0 k ) = )) x x k. k We sy tht the funtion f hs t the point strong grdient, symbolilly strgrd f( ), if for every k =,..., n there exist finite f [x k] (x0 ), nd we write strgrd f( ) = (f [x k ] (x0 ),..., f [x n ] (x0 )).
98 Bulletin of TICMI If there exists strgrd f( ), then there likewise exists nggrd f( ), nd equlities strgrd f( ) = nggrd f( ) = grd f( ) hold. Consequently, we hve Theorem 4. ([5]; [6]; [3, p. 77]) : The existene of finite strgrd f( ) implies existene of totl differentil df( ) nd strgrd f( ) = nggrd f( ) = grd f( ). If the grd f(x) is ontinuous t the point, then we hve the equlity strgrd f( ) = grd f( ) ([3, p. 75]). Besides, the existene of the finite strgrd f( ) does not, in generl, imply the ontinuity of grd f(x) t the point. For, the grdient of the funtion g(x, x 2 ) from the Proposition 3.5 is not ontinuous t (0, 0), but strgrd g(0, 0) = (0, 0). In ddition to this, we hve Theorem 4.2 ([3, p. 76]): There exists n bsolutely ontinuous funtion of two vribles whih hs lmost everywhere both finite strong nd disontinuous grdients. Remrk : We should begin the proof of Theorem 4.2 from [3] with the following. There exists mesurble set e [0, ] suh tht the sets e (α, β) nd ([0, ] \ e) (α, β) hve positive mesures for ll subintervls (α, β) [0, ] ([7, p. 50] or [8, p. 49]). By α(x) we denote the hrteristi funtion of the set e. Anlogously, we obtin the funtion β(y). Proposition 4.3 ([5]; [6]; [3, p. 77]) : The finiteness of nggrd f( ) or, wht is the sme, the existene of df( ) does not imply the existene of strgrd f( ). The funtion λ(x, x 2 ) = x x 2 2/3 is differentible t the point = (0.0) (see Proposition 4.3), but strgrd λ( ) does not exist. Afterwrds, G.G. Onini estblished tht the existene of finite strong grdient is essentilly stronger property thn the differentibility. Theorem 4.4 ([]; [2]) : f : [0, ] n R suh tht: For rbitrry n 2 there exists ontinuous funtion. f is differentible lmost everywhere, 2. f devoid of finite strong grdient lmost everywhere. The following theorem is improvement of Theorem 4.4. Theorem 4.5 ([0, Theorem 4]): For rbitrry n 2 there exists ontinuous funtion f : [0, ] n R tht is differentible lmost everywhere, but everywhere devoid of finite strong grdient. As is known, funtions of bounded vritionss in the Hrdy or Arzel sense possess the differentibility property lmost everywhere, i.e. hve finite ngulr grdients lmost everywhere. As to the existene of strong grdient, funtions of these lsses behve differently. Theorem 4.6 ([9]; [0]): Every funtion f : [0, ] n R of bounded vrition in the Hrdy sense hs finite strong grdient lmost everywhere.
Vol. 8, No., 204 99 Theorem 4.7 ([0, Theorem 3]): For rbitrry n 2 there exists ontinuous funtion f : [0, ] n R of bounded vrition in the Arzel sense tht everywhere devoid of finite the strong grdient. 5. Clssifition of funtions by vrious grdients Theorem 5. ([3, p. 80]): A lss with ontinuous t the point grdients of funtions is ontined stritly in lss with finite t the point strong grdients of funtions, nd the ltter is ontined stritly in lss of funtions with finite t the point ngulr grdients. This lss oinides with the lss of differentible t funtions. Remrk : The notions of ngulr nd strong grdients were generlized by Leri Bntsuri, who introdued the notion of grdient with respet to the bsis nd estblished, in prtiulr, the reltionship between the differentiblity nd the existene of the grdient whih he hs introdued ([3], [4]). 6. Differentibility of n indefinite integrl nd of n bsolutely ontinuous funtions Let the funtion of two vribles f be summble on the retngle Q = {(x, y) R 2 : x b, y d}, f L(Q). Consider for the funtion f the indefinite double integrl F (x, y) = x y f(t, τ) dt dτ. (6.) The following problems re quite nturl. I. Does the indefinite double integrl (6.), hve or hve no totl differentil lmost everywhere? II. If F hs totl differentil, then t wht points nd how the set of suh points is onneted with the funtion f? The nswer to problem I will be given here, nd problem II will be onsidered in Setion 8. Theorem 6. ([5, Theorem 6.7]; [5]; [3, pp. 02 04]): Indefinite integrl (6.) hs totl differentil t lmost ll points (x, y) Q for every funtion f L(Q). Theorem 6.2 ([5]; [5]; [3, p. 04]): At every point (, y 0 ) Q of differentibility of the indefinite integrl (6.) with f L(Q) we hve h 0 k 0 h + k x0 +h y0 +k y 0 f(t, τ) dt dτ = 0. (6.2) In prtiulr, equlity (6.2) is fulfilled t lmost ll points (, y 0 ) Q. Theorem 6.3 ([5]; [3, p. 05]): Let the indefinite integrl (6.) for f L(Q) hve in the neighborhood of the point (, y 0 ) Q finite F x, F y nd F x,y. Then for the funtion F to be differentible t the point (, y 0 ), it is neessry nd suffiient
00 Bulletin of TICMI tht h 0 k 0 h k h + k F x,y( + θ h, y 0 + θ 2 k) = 0, 0 < θ, θ 2 <. If, in ddition, F x,y = f in the neighborhood of the point (, y 0 ), then for the differentibility of the funtion F t (, y 0 ) it is neessry nd suffiient tht h 0 k 0 h k h + k f( + θ h, y 0 + θ 2 k) = 0, 0 < θ, θ 2 <. Theorem 6.4 ([5]; [5]; [3, pp. 07 08]): Every bsolutely ontinuous on the Q funtion hs totl differentil lmost everywhere on the Q. Its prtil nd mixed prtil derivtives re summble on the Q funtions. Note tht Theorem 6.4 is not true for seprtely bsolutely ontinuous funtions! 7. The finiteness of strong grdient of n indefinite integrl nd n bsolutely ontinuous funtion Theorem 7. ([5, Theorem 6.6]; [3, p. 09]): Let the funtion f be summble on the retngle Q = {(x, y) R 2 : x b, y d}. Then the orresponding indefinite integrl F (x, y) = possesses the following properties: x y f(t, τ) dt dτ ) for lmost every [, b] nd for every y 0 [, d] the F [x] (, y 0 ) is finite, nd F [x] (x, y) = y0 f(, τ) dτ; 2) for every [, b] nd for lmost every y 0 [, d] the F [y] (, y 0 ) is finite, nd F [y] (x, y) = x0 f(t, y 0 ) dt; 3) t lmost every point (, y 0 ) Q the strgrd F (, y 0 ) is finite. Theorem 7.2 ([5, Theorem 6.8]; [3, pp. 2]): For every summble on the retngle Q = {(x, y) R 2 : x b, y d} funtion f the following sttements re vlid: ) there exists mesurble set e [, b] with e = b, suh tht t every point (, y 0 ) with e nd y [, d] the integrl y 0 f(, τ) dτ is finite,
Vol. 8, No., 204 0 nd h 0 h y y 0 x0 +h y0 f(t, τ) dt dτ = y0 f(, τ) dτ; (7.) 2) there exists mesurble set e 2 [, d] with e 2 = d, suh tht t every point (, y 0 ) with [, b] nd y 0 e 2 the integrl f(t, y 0) dt is finite, nd y0 +k x0 x0 f(t, τ) dt dτ = f(t, y 0 ) dt; (7.2) k 0 k y x 0 3) equlities (7.) nd (7.2) re fulfilled simultneously t the points (, y 0 ) E, where E = e e 2, E = Q. To formulte this nd the subsequent theorems in short, we introdue the following mesurble sets: A) E = m( ), E = Q, e where the mesurble set e [, b] with e = b is dopted from sttement ) of Theorem 7.2, nd the vertil losed intervl m( ) is defined by the equlity B) E 2 = y 0 e 2 n(y 0 ), m( ) = {(, y) : y d}; E 2 = Q, where the mesurble set e 2 [, d] with e 2 = d is dopted from sttement 2) of Theorem 7.2, nd the horizontl losed intervl n(y 0 ) is defined by the equlity n(y 0 ) = {(x, y 0 ) : x b}. Now Theorem 7.2 n be rephrsed s follows. Theorem 7.3 ([3, p. 2]): For every funtion f L(Q), equlities (7.) nd (7.2) tke ple t the points (, y 0 ) E nd (, y 0 ) E 2, respetively. Equlities (7.) nd (7.2) re fulfilled simultneously t the points (, y 0 ) E 3, where E 3 = E E 2, E 3 = Q. Theorem 7.4 ([5]; [3, pp. 2 3]): sttements tke ple: For every funtion f L(Q) the following ) t the points (, y 0 ) E the equlity holds; (h,k) (0,0) h x0 +h y0 +k y 0 f(t, τ) dt dτ = 0 (7.3)
02 Bulletin of TICMI 2) t the points (, y 0 ) E 2 the equlity (h,k) (0,0) k x0+h y0+k y 0 f(t, τ) dt dτ = 0 (7.4) is vlid; 3) equlities (7.3) nd (7.4) re fulfilled simultneously t the points (, y 0 ) E 3, where E 3 = E E 2, E 3 = Q ; 4) t the points (, y 0 ) E 3 the equlity holds. h + k (h,k) (0,0) hk x0+h y0+k y 0 f(t, τ) dt dτ = 0 (7.5) Remrk ([3, p. 3]): If S(x, y) L(Q) is Sk s funtion, then the expression hk x+h y+k x y S(t, τ) dt dτ (7.6) hs the strong supper it + t every point (x, y) Q. At the sme time, Theorem 7.4 shows tht tending of expression (7.6) to + is subordinte to equlities (7.3) nd (7.4) t the points (, y 0 ) E E 2, i.e. hk x0+h y0+k y 0 ( S(t, τ) dt dτ = O mx(h, k) Theorem 7.5 ([3, pp. 3 4]): To every bsolutely ontinuous on the retngle Q funtion Φ there orresponds triple of funtions φ L(Q), g L([, b]) nd h L([, d]), suh tht the following sttements tke ple: ) for lmost every [, b] nd for every y 0 [, d] there exists the finite Φ [x] (, y 0 ), nd ). Φ [x] (, y 0 ) = y0 φ(, y) dy + g( ); 2) for every [, b] nd for lmost every y 0 [, d] there exists the finite Φ [y] (, y 0 ), nd Φ [y] (, y 0 ) = x0 φ(x, y 0 ) dx + h(y 0 ); 3) t lmost every point (, y 0 ) Q the strgrdφ(, y 0 ), Φ x,y(, y 0 ) nd Φ y,x(, y 0 ) re finite, nd Φ x,y(, y 0 ) = φ(, y 0 ) = Φ y,x(, y 0 ). An n 2-dimensionl nlogue of sttement 3) of Theorem 7. is
Vol. 8, No., 204 03 Theorem 7.6 ([6]; [7]; [3, p. ]): For every n 2 nd f L(0, ) n the indefinite integrl of f, t lmost every point is differentible, moreover, hs finite strong grdient. 8. Lebesgue s intense points nd finiteness t these points of strong grdient of n indefinite integrl Definition 8. ([5]; [3, p. 5]): Let the funtion f belong to the spe L p (Q) for some p. The point (, y 0 ) Q is lled jointly Lebesgue s intense point (of p-th degree) of the funtion f, symbolilly (, y 0 ) int L p x,y(f), if the following two onditions re fulfilled: (h,k) (0,0) h (h,k) (0,0) k x0 +h y0 +k y 0 y0 +k x0 +h y0 f(x, y) dy x0 f(x, y) dy p f(, y) dy dx = 0, (8.) p f(x, y 0 ) dx dy = 0. (8.2) When equlity (8.) is fulfilled, then the point (, y 0 ) is lled Lebesgue s intense point with respet to the vrible x (of p-th degree) of the funtion f, symbolilly (, y 0 ) int L p x(f). When equlity (8.2) is fulfilled, then the point (, y 0 ) is lled Lebesgue s intense point with respet to the vrible y (of p-th degree) of the funtion f, symbolilly (, y 0 ) int L p y(f). Theorem 8.2 ([3, p. 5]): Let the funtion f belong to the spe L p (Q) for some p. The following sttements tke ple: ) there exists mesurble set e [, b] with e = b, suh tht the set of ll points (, y 0 ) with e nd y 0 [, d] forms the set int L p x(f), int L p x(f) = Q ; 2) there exists mesurble set e 2 [, d] with e 2 = d, suh tht the set of ll points (, y 0 ) with [, b] nd y 0 e 2 forms the set int Lp y(f), int L p y(f) = Q ; 3) the set of ll points (, y 0 ) with e nd y 0 e 2 forms the set int L p x,y(f), int L p x,y(f) = Q. Let the funtion f L(Q). Then the orrespond- Theorem 8.3 ([5]; [3, p. 8]): ing indefinite integrl F (x, y) = possesses the following properties: x y f(t, τ) dt dτ ) t every point (, y 0 ) int L x (f) the F [x] (, y 0 ) is finite nd y0 F [x] (, y 0 ) = f(, τ) dτ,
04 Bulletin of TICMI or wht is the sme, h 0 h k 0 x0 +h y0 +k f(t, τ) dt dτ = y0 f(, τ) dτ; 2) t every point (, y 0 ) int L y (f) the F [y] (, y 0 ) is finite nd or wht is the sme, y0 +k h 0 k y k 0 0 x0 F [y] (, y 0 ) = f(t, y 0 ) dt, x0 +h f(t, τ) dt dτ = x0 f(t, y 0 ) dt; 3) t every point (, y 0 ) int L x,y (f) the strgrd F (, y 0 ) is finite, in prtiulr, there exists df (, y 0 ). Theorem 8.4 ([5]; [3, p. 9]): following sttements re vlid: For every funtion f L p (Q) with p the ) t every point (, y 0 ) int L p x(f) the equlity (h,k) (0,0) h x0 +h y0 +k holds; 2) t every point (, y 0 ) int L p y(f) we hve y0 +k (h,k) (0,0) k y 0 y 0 x0 +h p f(x, y) dy dx = 0 (8.3) p f(x, y) dx dy = 0; (8.4) 3) t every point (, y 0 ) int L p x,y(f) equlities (8.3) nd (8.4) re fulfilled simultneously. 9. A generliztion of lssil theorems on derivtives on n indefinite integrl nd of the definite integrl with prmeter We men the following Lebesgue s nd Ch. J. de l Vlle e Poussin s theorems. Theorem L: Let ψ be summble funtion on losed intervl [, b], nd suppose tht Ψ(x) = Then for lmost ll x [, b] the equlity x ψ(t) dt. Ψ (x) = ψ(x) (9.)
is fulfilled. Vol. 8, No., 204 05 Theorem VP: Let funtion f(x, y) be summble with respet to x on losed intervl [, b] for every fixed y from losed intervl [, d]. Consider finite on [, d] funtion definite integrl with prmeter y Φ(y) = b f(x, y) dx. Suppose tht the following onditions re fulfilled: (A) f(x, y) is funtion bsolutely ontinuous with respet to y on [, d] for every fixed x [, b]; (B) prtil derivtive f y with respet to y is summble funtion on losed retngle Q = {(x, y) R 2 : x b, y d}. Then for lmost ll y [, d]. Φ (y) = b f y(x, y) dx (9.2) We hve the following generliztion of Theorem L nd Theorem VP. Theorem 9. ([8]; [3, pp. 20 23]): Let ssumptions (A) nd (B) of Theorem VP be fulfilled. Then the funtion n indefinite integrl with prmeter y possesses the following properties: F (x, y) = x f(t, y) dt (9.3) (i) there exists e [, b] suh tht e = b, F [x] (, y 0 ) is finite for (, y 0 ) e [, d], nd F [x] (, y 0 ) = f(, y 0 ); (9.4) (ii) there exists e 2 [, d] suh tht e 2 = d, F [y] (, y 0 ) is finite for (, y 0 ) [, b] e 2, nd x0 F [y] (, y 0 ) = f y(t, y 0 ) dt; (9.5) (iii) the strgrd F (, y 0 ) is finite t lmost ll points (, y 0 ) Q, in prtiulr, there exists the totl differentil df (, y 0 ). If in equlity (9.3) we put x = b, then equlity (9.5) for = b tkes the form of equlity (9.2) beuse the derivtive of the funtion of one vrible is, in ft, its strong prtil derivtive with respet to the sme vrible, if we onsider this funtion s the funtion of two vribles, onstnt with respet to the seond vrible. Thus equlity (9.5) is the generliztion of equlity (9.2).
06 Bulletin of TICMI Equlity (9.) is obtined nlogously from equlity (9.4) if the funtion f in equlity (9.3) is ssumed to be independent of the vrible y. 0. A riterion of C n -differentibility Theorem 0. ([9]): nd only if the ondition A funtion f is C n -differentible t point z C n, if f x k (z) + if ŷ k (z) = 0 or, equivlently D xk f(z) + idŷk f(z) = 0 holds for ll k =,..., n, where z = (z,..., z n ) nd z k = x k + y k. Hrtog s Min Theorem([9, p. 7]). A funtion f holomorphi (nlyti) with respet to eh vrible in n open set G C n is C n -holomorphi (C n -nlyti) in G.. On the H-differentibility Definition. ([20]) : A quternion funtion f(z), z = + x i + x 2 i 2 + x 3 i 3, defined on some neighborhood of point z 0 = 0 + x0 i + 2 i 2 + 3 i 3, is lled H- differentible t z 0 if there exists two sequenes of quternions A k (z 0 ) nd B k (z 0 ) suh tht A k (z 0 )B k (z 0 ) is finite nd tht the inrement f(z 0 + h) f(z 0 ) of the k funtion f n be represented s f(z 0 + h) f(z 0 ) = k A k (z 0 )B k (z 0 ) + ω(z 0, h), where ω(z 0, h) = 0. h 0 h In this se, the quternion k A k (z 0 )B k (z 0 ) is lled the H-derivtive of the funtion f t the point z 0 nd is denoted f (z 0 ). Thus f (z 0 ) = k A k (z 0 )B k (z 0 ). (.) The uniqueness of the H-derivtive follows from the ft tht the right-hnd prt of (.), if it exists, is just the prtil derivtive f (z 0 ) of f t z 0 with respet to its rel vrible (see [6], equlity (2)). *For the history of differentibility of quternion funtions see [20] nd [2], p. 385.
Vol. 8, No., 204 07 The bsi elementry qunternion funtions z n, e z, os z, sin z re H- differentible nd fulfilled the following equlities (z n ) = nz n, (e z ) = e z, (os z) = sin z, (sin z) = os z. The rules for lulting H-derivtives re identil to those derived in stndrd lulus ourse: (f) (z) = f (z), (f) (z) = f (z), (f + φ) (z) = f (z)+φ (z), (fφ) (z) = f (z)φ(z)+f(z)φ (z), (/φ) (z) = /φ(z) φ (z) /φ(z), (f /φ) (z) == f (z) /φ(z) f(z) /φ(z) φ (z) /φ(z), (/φ f) (z) = /φ(z) φ (z) /φ(z) f(z) + /φ(z) f (z). Right now we formulte reltionship between H-differentibility of quternion funtion f(z) = u 0 (z) + u (z)i + u 2 (z)i 2 + u 3 (z)i 3 of quternion vrible z = + x i + x 2 i 2 + x 3 i 3 nd the existene of the differentil df(z) (with respet to rel vribles, x, x 2, x 3 ). Sine the prtil ngulr derivtives re the derivtives with respet to rel vribles (see the Setion 2), the ondition of differentibility for rel, omplex nd quternion funtions re expressed in the sme form. It then follows tht for the differentibility of quternion funtion f t point z = +x i +x 2 i 2 +x 3 i 3, neessry nd suffiient ondition is the existene finite prtil ngulr derivtives f x k = (u 0 ) x k +i (u 2 ) x k +i 2 (u 2 ) x k +i 3 (u 3 ) x k, k = 0,, 2, 3. Moreover, when f is differentible t z, the following equlities hold for its differentil df(z): df(z) = f (z)d + f x (z)dx + f x 2 (z)dx 2 + f x 3 (z)dx 3, df(z) = du 0 (z) + i du (z) + i 2 du 2 (z) + i 3 du 3 (z). Theorem.2 ([22]): If quternion funtion f is H-differentible t point z = + x i + x 2 i 2 + x 3 i 3, then f is differentible t the sme point z nd its prtil ngulr derivtives f (z), f x (z), f x 2 (z), f x 3 (z) n be expressed in terms of the H-derivtive f (z) = A k (z)b k (z) s follows: k f (z) = k A k (z)b k (z) = f (z), (.2) f x (z) = k A k (z)i B k (z), (.3) f x 2 (z) = k A k (z)i 2 B k (z), (.4) f x 3 (z) = k A k (z)i 3 B k (z). (.5) Moreover, we hve df(z) = k A k (z) dz B k (z). (.6) Remrk : Eqution (.6) n be interpreted s follows. As in the lssil se, the differentil df(z) of n H-differentible funtion f is liner with respet to the differentil dz of the independent vrible z.
08 Bulletin of TICMI Theorem.3 ([22]): If quternion funtion f is differentible t point z nd its prtil ngulr derivtives f (z), f x (z), f x 2 (z) nd f x 3 (z) n be expressed in the forms (.2) (.5) for some quternions A k (z) nd B k (z), then f is H- differentible t the point z nd f (z) = k A k (z)b k (z). We n ombine Theorems.2 nd.3 to obtin the following theorem Theorem.4 ([22]): The existene of the differentil df(z) of quternion funtion f nd its representbility in the form df(z) = k A k (z) dz B k (z) (.7) is equivlent to the existene of the derivtive f (z) nd its representbility in the form f (z) = k A k (z)b k (z). Corollry.5 ([22]): When x 2 = 0 = x 3 nd u 2 = 0 = u 3, then one hs omplex funtion f(z) = u(z) + iv(z) of omplex vrible z = x + iy. In this se, Eq. (.7) hs the form df(z) = (z)dz = (z)dx + i(z)dy, where (z) = k A k(z)b k (z), from whih we obtin the equlities f x (z) = (z) nd f ŷ (z) = i(z). Thus, we hve f x (z) + if ŷ (z) = 0. (.8) Note tht Eq.(.8) is neessry nd suffiient ondition for the omplex funtion f to be C -differentible t the point z (see [9], Theorem 3., when n = ). Moreover, we hve obtined the well known equlities f (z) = f x (z) nd f (z) = if ŷ (z) for the derivtive f (z). Corollry.6 ([22]): For quternion z = + x i + x 2 i 2 + x 3 i 3, we hve dz n = z n dz + z n 2 dz z + z n 3 dz z 2 + + zdz z n 2 + dz z n for ll n = 0,, 2,.... Corollry.7 ([22]): For the prtil derivtives of the funtions f n (z) = z n, n = 0,, 2,..., with respet to rel vribles x k, k = 0,, 2, 3, we hve (z n ) x k = z n i k + z n 2 i k z + + z i k z n 2 + i k z n. Referenes [] G.P. Tolstov, On prtil derivtives (Russin), Izvestiy Akd. Nuk SSSR. Ser. Mt., 3 (949), 425-446; trnsltion in Amer. Mth. So. (952), 55-83 [2] O.P. Dzgnidze, Seprtely ontinuous funtions in new sense re ontinuous, Rel Anl. Exhnge, 24, 2 (998/99), 695-702 [3] O. Dzgnidze, Some new results on the ontinuity nd differentibility of funtions of severl rel vribles, Pro. A. Rzmdze Mth. Inst., 34 (2004), -38 [4] Z. Piotrowski, The genesis of seprte versus joint ontinuity, Ttr Mt. Mth. Publ., 8 (996), 3-26
Vol. 8, No., 204 09 [5] O.P. Dzgnidze, On the differentibility of funtions of two vribles nd of indefinite double integrls, Pro. A. Rzmdze Mth. Inst., 06 (993), 7-48 [6] O. Dzgnidze, A neessry nd suffiient ondition for differentibility funtions of severl vribles, Pro. A. Rzmdze Mth. Inst., 23 (2000), 23-29 [7] Yu.S. Očn, Colletion of Problems nd Theorems on the Theory of Funtions of Rel Vrible, Prosveshheniye, Mosow, 965 [8] Yu.S. Očn, Colletion of Problems on Mthemtil Anlysis (generl theory of sets nd funtions), Prosveshheniye, Mosow, 98 [9] L.D. Bntsuri, G.G. Onini, On the differentil properties of funtions of bounded vrition in Hrdy sense, Pro. A. Rzmdze Mth. Inst., 39 (2005), 93-95 [0] L.D. Bntsuri, G.G. Onini, On differentil properties of funtions of bounded vrition, Anl. Mth., 38, (202), -7 [] G. Onini, On the reltion between onditions of the differentibility nd existene of the strong grdient, Pro. A. Rzmdze Mth. Inst., 32 (2003), 5-52 [2] G.G.Onini, On the inter-reltion between differentibility onditions nd the existene of strong grdient (Russin), Mt. Zmetki, 77, (2005), 93-98; trnsltion in Mth. Notes, 77, -2 (2005), 84-89 [3] L. Bntsuri, On the reltion between the differentibility ondition nd the ondition of the existene of generlized grdient, Bull. Georgin Ad. Si., 7, 2 (2005), 24-242 [4] L. Bntsuri, On the reltionship between the onditions of differentibility nd existene of generlized grdient, Book of Abstrts of the III Interntionl Conferene of the Georgin Methemtil Union (Btumi, 202, September 2-9), 65-66 [5] O. Dzgnidze, Totl differentil of the indefinite Lebesgue integrl, Pro. A. Rzmdze Mth. Inst., 4 (997), 27-34 [6] O. Dzgnidze, G. Onini, On one nlogue of Lebesgue theorem on the differentition of indefinite integrl for funtions of severl vribles, Pro. A. Rzmdze Mth. Inst., 32 (2003), 39-40 [7] O. Dzgnidze, G. Onini, On one nlogue of Lebesgue theorem on the differentition of indefinite integrl for funtions of severl vribles, Pro. A. Rzmdze Mth. Inst., 33 (2003), -5 [8] O. Dzgnidze, The Lebesgue nd de l Vllée Poussin s theorems on derivtion of n integrl, Ttr Mt. Mth. Publ., 35 (2007), 07-3 [9] O. Dzgnidze, A riterion of joint C-differentibility nd new proof of Hrtogs min theorem, J. Appl. Anl., 3, (2007), 3-7 [20] O. Dzgnidze, On the differentibility of quternion funtions, Tbil. Mth. J., 5 (202), -5. ArXiv: 203.569V[mt.CV] 26 Mr, 202 [2] G.E. Shilov, Mthemtil Anlysis: Funtions of Severl Rel Vribles, prts 2. Nuk, Hed Editoril Offie for Physil nd Mthemtil Literture, Mosow, 972. [22] O. Dzgnidze, Neessry nd suffiient onditions for H-differentibility of quternion funtions, Georgin Mth. J. (submitted).