CONTENTS 0. Introduction and notations...................................................5 1. Basic properties of delta-convex mappings.........................8 2. Delta-convex curves..........................................................15 3. Differentiability of delta-convex mappings.........................17 A. First derivative...............................................................17 B. Second derivative of mappings $F: R^n → Y$...............23 4. Superpositions and inverse mappings..............................26 5. Inverse mappings in finite-dimensional case.....................31 6. Examples and applications................................................34 A. Three counterexamples.................................................34 B. Nemyckii and Hammerstein operators............................36 C. Weak solution of a differential equation.........................38 D. Quasidifferentiable functions and mappings..................41 7. Some open problems........................................................44 References...........................................................................47
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On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.
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