316 |
Jerison M., Rings of Continuous Functions
- Gillman
- 1960
(Show Context)
Citation Context ...e f ′iα (x+ td; d) = sup0≤t≤tα f ′ iα (x+ td; d) a.e.. 5The stronger compactness assumption is necessary: There exists some countably compact space X whose square X ×X is not even pseudocompact, see [=-=Gillman and Jerison, 1960-=-][page 135, Example 9.15]. Augmenting with [Engelking, 1989][page 238, Problem 3.12.21] we know there exists a continuous function f : X ×X 7→ R whose pointwise supremum over the countably compact spa... |

211 | Envelope theorems for arbitrary choice sets. - Milgrom, Segal - 2002 |

154 | Convex Analysis and Minimization Algorithms, volume I and II. - Hiriart-Urruty, Lemarechal - 1996 |

58 |
Optima and Equilibria: An Introduction to Nonlinear Analysis,
- Aubin
- 1998
(Show Context)
Citation Context ...the l.s.c. of f ′(x; ·) (being a pointwise supremum of continuous functions f ′i(x; ·), whose continuity is guaranteed by the continuity of fi at x). Remark 5 The above beautiful proof is taken from [=-=Aubin, 1998-=-], see also [Hiriart-Urruty and Lemaréchal, 1993]. Comparing Theorem 1 and 3 we see that the extra convexity assumption dispenses the assumptions on the directional derivatives, which justifies our s... |

8 |
On a theorem of Danskin with an application to a theorem of von Neumann-Sion
- Bernhard, Rapaport
- 1995
(Show Context)
Citation Context ...n the proof of Theorem 1 that f(x+ tnd) <∞ for all |t| sufficiently small. 4 Remark 3 Had f := infi∈I fi, we only need to change assumption 2 to infi∈I f ′i(x; d) > −∞. Theorem 2 was proved first by [=-=Bernhard and Rapaport, 1995-=-] under some unnecessary assumptions. Our treatment here combines some idea presented in [Milgrom and Segal, 2002]. Note that Corollary 1 also follows from Theorem 2: the joint continuity of ∇xf(x, y)... |

4 |
General Topology. Heldermann Verlag Berlin, revised and completed edition
- Engelking
- 1989
(Show Context)
Citation Context ...compactness assumption is necessary: There exists some countably compact space X whose square X ×X is not even pseudocompact, see [Gillman and Jerison, 1960][page 135, Example 9.15]. Augmenting with [=-=Engelking, 1989-=-][page 238, Problem 3.12.21] we know there exists a continuous function f : X ×X 7→ R whose pointwise supremum over the countably compact space X is not continuous. This argument is due to AliReza Olf... |

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