summary:Zaj'\i ček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any real-valued function on a normed linear space, the set of points where the function is Fréchet directionally differentiable but not Fréchet differentiable is first category

Giles, J. R.

Sciffer, Scott

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Fréchet directional differentiability and Fréchet differentiability

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeJohn R. Giles; Scott ScifferFréchet directional differentiability and Fréchet differentiabilityCommentationes Mathematicae Universitatis Carolinae, Vol. 37 (1996), No. 3, 489--497Persistent URL: http://dml.cz/dmlcz/118855Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1996Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 37,3 (1996)489–497 489Fréchet directional differentiabilityand Fréchet differentiabilityJ.R. Giles, Scott ScifferAbstract. Zaj́ıček has recently shown that for a lower semi-continuous real-valued func-tion on an Asplund space, the set of points where the function is Fréchet subdifferentiablebut not Fréchet differentiable is first category. We introduce another variant of Fréchetdifferentiability, called Fréchet directional differentiability, and show that for any real-valued function on a normed linear space, the set of points where the function is Fréchetdirectionally differentiable but not Fréchet differentiable is first category.Keywords: Gâteaux and Fréchet subdifferentiability, directional differentiability, strictand intermediate differentiabilityClassification: 58C20, 46G05A real-valued function F on an open subset A of a normed linear space Xis said to be Fréchet differentiable at x ∈ A if there exists a continuous linearfunctional F ′(x) on X where, given ǫ > 0 there exists a δ(ǫ, x) > 0 such that|F (x+ y)− F (x)− F ′(x)(y)| < ǫ‖y‖ for all y ∈ X, ‖y‖ < δ.In determining Fréchet differentiability properties, interest has focused on variantsof Fréchet differentiability. The function F is said to be Fréchet subdifferentiableat x ∈ A if there exists a continuous linear functional f on X where, given ǫ > 0there exists a δ(ǫ, x) > 0 such thatF (x+ y)− F (x)− f(y) > −ǫ‖y‖ for all y ∈ X, ‖y‖ < δ.In particular, Borwein and Preiss proved that when X is a Banach space withan equivalent norm Fréchet differentiable away from the origin, a lower semi-continuous function F on an open subset A of X is densely Fréchet subdifferen-tiable, [BP, p. 521]. Recently Zaj́ıček proved that when X is an Asplund space, alower semi-continuous function F on an open subset A of X has the property thatthe set of points where F is Fréchet subdifferentiable but not Fréchet differentiableis first category in A, [Z2, p. 485].In this paper we study another variant of Fréchet differentiability. Given areal-valued function F on an open subset A of a normed linear space X , we saythat F has a right-hand derivative at x ∈ A in the direction v ∈ X ifF ′+(x)(v) = limλ→0+F (x+ λv)− F (x)λ490 J.R.Giles, Scott Scifferexists. Clearly F ′+(x)(v) is positively homogeneous in v. We say F is directionallydifferentiable at x ∈ A if F ′+(x)(v) exists in every direction v ∈ X and is acontinuous function in v. If F ′+(x)(v) is also linear in v then we say that F isGâteaux differentiable at x. We note that although F may have a right-handderivative at x ∈ A in every direction v ∈ X , F ′+(x)(v) need not be continuous inv, even if F is continuous at x.Example. Consider F on R2 defined in polar coordinates byF (r, θ) ={cos θ sin(r/ cos θ) for cos θ 6= 0 and r 6= 00 for cos θ = 0 or r = 0.Now at the origin∂F∂r={1 when cos θ 6= 00 when cos θ = 0.However, if a locally Lipschitz function has a right-hand derivative at a pointin every direction, then it is directionally differentiable at the point. This is theimplication of the following well known result whose proof is included for the sakeof completeness.Proposition 1. Consider a locally Lipschitz function ψ on an open subset A ofa normed linear space X . Given x ∈ A, if ψ′+(x)(v) exists for all v ∈ X thenψ′+(x)(v) is Lipschitz in v.Proof: Since ψ is locally Lipschitz on A, there exists a K > 0 and a δ > 0 suchthat|ψ(y)− ψ(z)| ≤ K‖y − z‖ for all y, z ∈ B(x; δ) ∩A.Given u ∈ X , ‖u‖ ≤ 1 and ǫ > 0 there exists a 0 < δ1 < δ such thatψ′+(x)(u)− ǫ <ψ(x + λu)− ψ(x)λfor 0 < λ < δ1≤ψ(x + λv)− ψ(x)λ+K‖u− v‖for v ∈ X, ‖v‖ ≤ 1 and 0 < λ < δ1.But there exists 0 < δ2 < δ1 such thatψ(x+ λv)− ψ(x)λ< ψ′+(x)(v) + ǫ for 0 < λ < δ2.And soψ′+(x)(u) − ǫ < ψ′+(x)(v) + ǫ+K‖u− v‖ for all u, v ∈ X, ‖u‖, ‖v‖ ≤ 1.Fréchet directional differentiability and Fréchet differentiability 491This holds for all ǫ > 0 and so we conclude thatψ′+(x)(u) ≤ ψ′+(x)(v) +K‖u− v‖and so|ψ′+(x)(u)− ψ′+(x)(v)| ≤ K‖u− v‖ for all u, v ∈ X,since ψ′+(x)(v) is positively homogeneous in v. For a real-valued function F on an open subset A of a normed linear space Xwe say that F is Fréchet directionally differentiable at x ∈ A if F is directionallydifferentiable at x and given ǫ > 0 there exists δ(ǫ, x) > 0 such that∣∣∣∣F (x + λv)− F (x)λ− F ′+(x)(v)∣∣∣∣< ǫ for all 0 < λ < δ and all v ∈ X, ‖v‖ ≤ 1.Of course F is Fréchet differentiable at x ∈ A if it is Fréchet directionally differ-entiable at x and F ′+(x)(v) is linear in v.It is of interest to see how Fréchet directional differentiability and Fréchetsubdifferentiability relate. A continuous convex function φ on an open convexsubset A of a normed linear spaceX has a subgradient at each point of its domain;that is, given x ∈ A there exists a continuous linear functional f on X such thatφ(x + y)− φ(x) ≥ f(y) for all y ∈ X, [Ph, p. 7].So φ is Fréchet subdifferentiable at each point in its domain. But also φ′+(x)(y)exists and is a continuous sublinear functional in y at each point x ∈ A, [Ph,p. 2], so φ is directionally differentiable at each point of its domain. However, thenorm is always Fréchet directionally differentiable at the origin, but it need notbe Fréchet directionally differentiable at any other point; there exists on ℓ1 anequivalent norm which is Gâteaux differentiable away from the origin but whichis nowhere Fréchet differentiable, [Ph, p. 86].More generally, for a real-valued function F on an open subset A of a normedlinear space X , if F is Fréchet directionally differentiable at x ∈ A then givenǫ > 0 there exists a δ > 0 such thatF (x+ y)− F (x) > F ′+(x)(y)− ǫ‖y‖ for all y ∈ X, ‖y‖ < δ,and if F ′+(x)(y) is also sublinear in y then any subgradient f of F′+(x) at x satisfiesF (x + y)− F (x) > f(y)− ǫ‖y‖ for all y ∈ X, ‖y‖ < δ,that is, F is Fréchet subdifferentiable at x. However, if F is Fréchet directionallydifferentiable at x then it is not necessarily Fréchet subdifferentiable at x; for thenorm ‖.‖ on X , F = −‖.‖ is Fréchet directionally differentiable at 0 but is notFréchet subdifferentiable at 0.We now establish for Fréchet directional differentiability a general result com-parable to that of Zaj́ıček for Fréchet subdifferentiability, [Z2, p. 485].492 J.R.Giles, Scott ScifferTheorem 2. Given a real-valued function F on an open subset A of a normedlinear space X , the set of points W ⊂ A where F is Fréchet directionally differ-entiable, but not Fréchet differentiable, is first category in A.Proof: For each n, p ∈ N consider the set Wn,p consisting of those points in Awhere F is Fréchet directionally differentiable and(i)∣∣∣∣F (x+ λv) − F (x)λ− F ′+(x)(v)∣∣∣∣<1pfor all 0 < λ ≤2nand all v ∈ X, ‖v‖ ≤ 1,(ii)there exist u, v ∈ X, ‖u‖, ‖v‖ ≤ 1 and|F ′+(x)(u) + F′+(x)(v) − F′+(x)(u + v)| >16p.NowW = ∪n,p≥2Wn,p and we show that for each n, p ≥ 2, Wn,p is nowhere densein A.Suppose on the contrary that for some n, p ≥ 2, Wn,p is dense in some opensubset U of A. We may assume that 0 ∈ Wn,p ∩ U and F (0) = 0. We examineestimates of F near 0. For m > n, by (i), we have|F (1nu)−1nF ′+(0)(u)| <1np, |F (1mv)−1mF ′+(0)(v)| <1mpand|F (1m(u + v))−1mF ′+(0)(u + v)| <1mp.We now choose m sufficiently large that B(0; 2/m) ⊂ U and, to satisfy the conti-nuity of F ′+(0), such that|F ′+(0)(1nu)− F ′+(0)(x+1nu)| <1npfor all x ∈ B(0; 2/m).Since Wn,p is dense in U , by the continuity of F′+(0), there exists an xm ∈B(0; 2/m) ∩Wn,p such that both|F ′+(0)(1mv)− F ′+(0)(xm)| <1mpand|F ′+(0)(1m(u+ v))− F ′+(0)(xm +1mu)| <1mp.However, by property (i), |F (xm)− F′+(0)(xm)| < (1/p)‖xm‖ and so|F (xm)− F′+(0)(1mv)| ≤ |F (xm)− F′+(0)(xm)|+ |F′+(0)(xm)− F′+(0)(1mv)|<1p‖xm‖+1mp<3mp. . . . (a)Fréchet directional differentiability and Fréchet differentiability 493Similarly, when m ≥ 2n we have|F (xm +1mu)− F ′+(0)(1m(u+ v))|≤ |F (xm +1mu)− F ′+(0)(xm +1mu)|+ |F ′+(0)(xm +1mu)− F ′+(0)(1m(u + v))|<1p‖xm +1mu‖+1mp<4mpsince ‖xm +1mu‖ ≤3m≤2n. . . . (b)Also|F (xm +1nu)− F ′+(0)(1nu)|≤ |F (xm +1nu)− F ′+(0)(xm +1nu)|+ |F ′+(0)(xm +1nu)− F ′+(0)(1nu)|<1p‖xm +1nu‖+1np<3np. . . . (c)Since xm ∈Wn,p,∣∣∣∣F (xm + (1/m)u)− F (xm)1/m−F (xm + (1/n)u)− F (xm)1/n∣∣∣∣≤∣∣∣∣F (xm + (1/m)u)− F (xm)1/m− F ′+(xm)(u)∣∣∣∣+∣∣∣∣F (xm + (1/n)u)− F (xm)1/n− F ′+(xm)(u)∣∣∣∣<2p.But∣∣∣∣F (xm + (1/m)u)− F (xm)1/m−F (xm + (1/n)u)− F (xm)1/n∣∣∣∣≥∣∣∣∣F ′+(0)((1/m)(u + v))− F′+(0)((1/m)v)1/m−F ′+(0)((1/n)u)− F′+(0)((1/m)v)1/n∣∣∣∣−m|F ′+(0)((1/m)(u + v))− F (xm + (1/m)u)| −m|F′+(0)((1/m)v)− F (xm)|− n|F ′+(0)((1/n)u)− F (xm + (1/n)u)| − n|F′+(0)((1/m)v)− F (xm)|≥ |F ′+(0)(u+ v)− F′+(0)(v)− F′+(0)(u)| −nm|F ′+(0)(v)| −4p−3p−3p−3nmpfrom (a), (b) and (c)≥16p−nm|F ′+(0)(v)| −12p≥3pfor any choice of m ≥ np|F ′+(0)(v)|. This is our contradiction, so we concludethat W is first category. 494 J.R.Giles, Scott ScifferA real-valued function F on an open subset A of a normed linear space X issaid to be strictly differentiable at x ∈ A iflimz→xλ→0+F (z + λv)− F (z)λexists for all v ∈ Xand this limit is F ′(x)(v), where F ′(x) is a continuous linear functional on X .Further, F is said to be uniformly strictly differentiable at x ∈ A if this limitexists and is approached uniformly for all v ∈ X , ‖v‖ ≤ 1.For a real-valued function on an open subset of a normed linear space the setof points where the function is Fréchet differentiable but not uniformly strictlydifferentiable is first category in its domain, [Z1, p. 158]. So using this fact we canmake the following deduction.Corollary 2.1. Given a real-valued function F on an open subset A of a normedlinear space X , the set of points where F is Fréchet directionally differentiablebut not uniformly strictly differentiable is first category in A.For a real-valued function on an open subset of a normed linear space theset of points where the function is both Fréchet subdifferentiable and Fréchet di-rectionally differentiable contains the set of points where the function is Fréchetdifferentiable. So from Theorem 2 and Zaj́ıček’s result for Fréchet subdifferentia-bility, [Z2, p. 485], we can produce the following relation between points of Fréchetsubdifferentiability and Fréchet directional differentiability.Corollary 2.2. Consider a real-valued function F on an open subset A of anormed linear space X .(i) The set of points where F is Fréchet directionally differentiable but notFréchet subdifferentiable is first category in A.(ii) If F is lower semi-continuous on A and X is an Asplund space thenthe set of points where F is Fréchet subdifferentiable but not Fréchetdirectionally differentiable is first category in A.We pointed out before Theorem 2 that there exists on ℓ1 an equivalent normwhich is Gâteaux differentiable away from the origin but is nowhere Fréchet dif-ferentiable. Now this norm is everywhere Fréchet subdifferentiable but Fréchetdirectionally differentiable only at the origin, so there is no obvious improvementto be made to Corollary 2.2 (ii). In fact for any Banach space X which is notan Asplund space there exists a continuous convex function φ on an open convexsubset A of X where the set of points of Fréchet differentiability is first categoryin A. So the set of points where φ is Fréchet subdifferentiable but not Fréchetdifferentiable is residual in A and from Theorem 2, the set of points where φ isFréchet subdifferentiable but not Fréchet directionally differentiable is residualin A. We should point out that for Lipschitz functions on the real line it may bethat the set of points of Fréchet subdifferentiability is first category; there is a Lip-schitz function ψ where the set of points of differentiability is first category, [GS2,Fréchet directional differentiability and Fréchet differentiability 495p. 210], so it is not possible that both ψ and −ψ are Fréchet subdifferentiable ona residual set.For convex functions there is a useful continuity characterization of Fréchetdirectional differentiability which has had interesting geometrical consequences.Given a continuous convex function φ on an open convex subset A of a normedlinear space X , the subdifferential of φ at x is the set∂φ(x) = {f ∈ X∗ : f(y) ≤ φ′+(x)(y) for all y ∈ X}.We say that the subdifferential mapping x 7→ ∂φ(x) is restricted norm uppersemi-continuous at x ∈ A if given ǫ > 0 there exists a δ > 0 such that∂φ(y) ⊂ ∂φ(x) + ǫB(X) for all y ∈ B(x; δ).The function φ is Fréchet directionally differentiable at x ∈ A if and only if thesubdifferential mapping x 7→ ∂φ(x) is restricted norm upper semi-continuous atx, [GM, Theorem 3.2]. It has been shown that a Banach space X is an Asplundspace if it has an equivalent norm which is Fréchet directionally differentiable onthe unit sphere, [CP, p. 453].It is instructive to see that there is no comparable continuity characterization ofFréchet directional differentiability for locally Lipschitz functions. Given a locallyLipschitz function ψ on an open subset A of a normed linear space X , the Clarkesubdifferential of ψ at x ∈ A is the set∂ψ(x) = {f ∈ X∗ : f(y) ≤ lim supz→xλ→0+ψ(z + λy)− ψ(z)λfor all y ∈ X}.Now ψ is strictly differentiable at x ∈ A if and only if ∂ψ(x) is singleton andis uniformly strictly differentiable at x ∈ A if and only if the subdifferentialmapping x 7→ ∂ψ(x) is single-valued and norm upper semi-continuous at x, [GS1,p. 374]. There exists a locally Lipschitz function which is Fréchet differentiable andstrictly differentiable at a point but not uniformly strictly differentiable at thatpoint, [GS1, p. 373]; so the function is Fréchet directionally differentiable at thepoint but its subdifferential is not restricted norm upper semi-continuous there.But also there exists a Lipschitz function on the real line whose subdifferentialmapping is everywhere restricted norm upper semi-continuous but nowhere single-valued, [GS2, p. 210], so by Corollary 2.1 the set of points where the function isnot Fréchet directionally differentiable is residual.We might well ask how our Theorem 2 generalizes for Gâteaux differentiability.For locally Lipschitz functions on certain Banach spaces we have sufficient infor-mation about generic sets of points of differentiability to achieve a generalization.Given a real-valued function F on an open subset A of a normed linear space Xwe say that F is intermediately differentiable at x ∈ A if there exists a continuouslinear functional f on X such thatlim infλ→0+F (x+ λv) − F (x)λ≤ f(v) ≤ lim supλ→0+F (x+ λv) − F (x)λfor all v ∈ X.496 J.R.Giles, Scott ScifferClearly if F ′+(x)(v) exists at x ∈ A for all v ∈ X and F is intermediately differ-entiable at x then F is Gâteaux differentiable at x.A GSG space is a Banach space which contains a dense continuous linearimage of an Asplund space. Every closed linear subspace of a GSG space is aweak Asplund space. It has been shown that a locally Lipschitz function ψ onan open subset A of a closed linear subspace X of a GSG space is intermediatelydifferentiable on a residual subset of A, [FP, p. 733]. So we can make an immediatededuction.Theorem 3. For a locally Lipschitz function ψ on an open subset A of a closedlinear subspace X of a GSG space, the set{x ∈ A : ψ′+(x)(v) exists for all v ∈ X andψ is not Gâteaux differentiable at x}is first category in A.For locally Lipschitz functions on separable Banach spaces rather more can bestated. For a locally Lipschitz function ψ on an open subset A of a separableBanach space X the set{x ∈ A : ψ is Gâteaux differentiable at x andψ is not strictly differentiable at x}is first category in A, [GS2, p. 210]. So from Theorem 3 we deduce the extendedresult.Theorem 4. For a locally Lipschitz function ψ on an open subsetA of a separableBanach space, the set{x ∈ A : ψ′+(x)(v) exists for all v ∈ X andψ is not strictly differentiable at x}is first category in A.We should make the following remarks about these results. Theorem 3 does nothold generally for all Banach spaces; on ℓ∞ the continuous semi-norm p definedfor x = (x1, x2, . . . , xn, . . . ) by p(x) = lim supn→∞ |xn| has p′+(x)(v) existingat each x ∈ ℓ∞ for all v ∈ ℓ∞, but p is nowhere Gâteaux differentiable, [Ph,p. 13]. Even for a locally Lipschitz function on an open interval of the real line,Theorem 4 has no obvious improvement; there exists a locally Lipschitz functionψ everywhere differentiable on (a, b) and which is strictly differentiable only on aset of less than full measure, [M, p. 975].Fréchet directional differentiability and Fréchet differentiability 497References[BP] Borwein J.M., Preiss D., A smooth variational principle with applications to subdifferen-tiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987),517–527.[CP] Contreras M.D., Paya R., On upper semi-continuity of duality mappings, Proc. Amer.Math. Soc. 121 (1994), 451–459.[FP] Fabian M., Preiss D., On intermediate differentiability of Lipschitz functions on certainspaces, Proc. Amer. Math. Soc. 113 (1991), 733–740.[GM] Giles J.R., Moors W.B., Generic continuity of restricted weak upper semi-continuousset-valued mappings, Set-valued Analysis, to appear.[GS1] Giles J.R., Scott Sciffer, Continuity characterizations of differentiability of locally Lip-schitz functions, Bull. Austral. Math. Soc. 41 (1990), 371–380.[GS2] Giles J.R., Scott Sciffer, Locally Lipschitz functions are generically pseudo-regular onseparable Banach spaces, Bull. Austral. Math. Soc. 47 (1993), 203–210.[M] Marcus S., Sur les fonctions dérivées, intégrables, au sens de Riemann et sur les dérivéespartialles mixtes, Proc. Amer. Math. Soc. 9 (1958), 973–978.[Ph] Phelps R.R., Convex Functions, Monotone Operators and Differentiability, Springer-Verlag, Lecture Notes in Math. 1364, 2nd ed., 1993.[Z1] Zaj́ıček L., Strict differentiability via differentiability, Act. U. Carol. 28 (1987), 157–159.[Z2] Zaj́ıček L., Fréchet differentiability, strict differentiability and subdifferentiability, Czech.Math. J. 41 (1991), 471–489.The University of Newcastle, NSW 2308, Australia(Received May 5, 1995)

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Fréchet directional differentiability and Fréchet differentiability

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