PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 1, January 2015, Pages 129–139 S 0002-9939(2014)12206-4 Article electronically published on August 15, 2014 CONSTRUCTION OF PATHOLOGICAL G ˆ ATEAUX DIFFERENTIABLE FUNCTIONS ROBERT DEVILLE, MILEN IVANOV, AND SEBASTI ´ AN LAJARA (Communicated by Thomas Schlumprecht) Abstract. We prove that for many pairs (X, Y ) of classical Banach spaces, there exists a bounded, Lipschitz, Gˆateaux differentiable function from X to Y whose derivatives are all far apart. 1. Introduction Let F be a function between real Banach spaces X and Y . We say that F has the jump property if F is Gˆateaux differentiable at every point of X and there exists a constant α> 0 such that ‖F ′ (x) − F ′ (y)‖≥ α whenever x, y ∈ X and x = y. We say that the couple (X, Y ) has the jump property if there exists a Lipschitz continuous, bounded function F : X −→ Y with the jump property. This concept was first considered by Deville and H´ajekin [9], where it was shown that the couple (X, R) never has the jump property and that such a property cannot be achieved if we replace Gˆ ateaux by Fr´ echet differentiability. There, it was also proved that (ℓ 1 , R 2 ) has the jump property and that if 1 ≤ p, q < ∞, then (ℓ p ,ℓ q ) enjoys it if and only if p ≤ q. Later on, Bayart [5] proved that if X is any separable infinite dimensional Banach space, then (X, c 0 ) has the jump property. Notice that a couple of Banach spaces (X, Y ) has the jump property if and only if there exists a Lipschitz continuous, bounded and Gˆateaux differentiable function F : X −→ Y such that ‖F ′ (x) − F ′ (y)‖≥ 1 whenever x and y are different elements of X. It is also clear that if the couple (X, Y ) has the jump property, then the space L(X, Y ) of bounded linear operators from X into Y is nonseparable, and that if Z is a Banach space that contains an isomorphic copy of Y , then the couple (X, Z ) has the jump property as well. A rather opposite kind of construction was provided in [3], where it was shown that if X and Y are separable Banach spaces, then there exists a continuous Gˆ ateaux differentiable function F : X −→ Y such that F ′ (X)= L(X, Y ). Some more results in this direction, in the case of Fr´ echet differentiability, were obtained in [4], [6], [10] and [11]. Received by the editors July 10, 2012 and, in revised form, January 14, 2013 and February 13, 2013. 2010 Mathematics Subject Classification. Primary 46B20, 46G05; Secondary 46T20. The second author was partially supported by NIS-SU, contract No. 133/2012. The third author was partialy supported by MTM2011-25377 (Ministerio de Ciencia e Inno- vaci´on) and by JCCM PEII11-0132-7661. c 2014 American Mathematical Society Reverts to public domain 28 years from publication 129 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use