Springer Science & Business Media, Mar 2, 2009 - Science - 461 pages
"An elegantly written, introductory overview of the field, with a near perfect choice of what to include and what not, enlivened in places by historical tidbits and made eminently readable throughout by crisp language. It has succeeded in doing the near-impossible—it has made a subject which is generally inhospitable to nonspecialists because of its ‘family jargon’ appear nonintimidating even to a beginning graduate student."
—The Journal of the Indian Institute of Science
"The book under review gives a comprehensive treatment of basically everything in mathematics that can be named multivalued/set-valued analysis. It includes...results with many historical comments giving the reader a sound perspective to look at the subject...The book is highly recommended for mathematicians and graduate students who will find here a very comprehensive treatment of set-valued analysis."
"I recommend this book as one to dig into with considerable pleasure when one already knows the subject...‘Set-Valued Analysis’ goes a long way toward providing a much needed basic resource on the subject."
—Bulletin of the American Mathematical Society
"This book provides a thorough introduction to multivalued or set-valued analysis...Examples in many branches of mathematics, given in the introduction, prevail [upon] the reader the indispensability [of dealing] with sequences of sets and set-valued maps...The style is lively and vigorous, the relevant historical comments and suggestive overviews increase the interest for this work...Graduate students and mathematicians of every persuasion will welcome this unparalleled guide to set-valued analysis."
It is said to be upper semicontinuous if and only if it is upper semi- continuous at any point ofDom(F). When F(x) is compact, F is upper semicontinuous at x if and only V e >0, 3 17 >0 such that V x' € Bx(x,n), F(x') C BY(F(x),e) We ...
Figure 1.3: Semicontinuous and Noncontinuous Maps is upper semicontinuous at zero but not lower semicontinuous at zero. □ We are therefore led to introduce still another Definition 1.4.3 We shall say that set-valued map F is continuous ...
F(a;/) Then we can measure the lack of closedness (of the graph) or the lack of lower semicontinuity by the discrepancy ... The graph of an upper semicontinuous set-valued map F : X with closed domain and closed values is closed.
Corollary 1.4.10 Let F : X be a closed set-valued map and be an upper semicontinuous function. If the dimension of Y is finite , then the cut set- valued map Fr : X ~> V defined by is upper semicontinuous. This is due to the fact that ...
F is upper semicontinuous 2. — F-1 is proper (and thus, its domain is closed) Proof — Since for all x S Dom(F), there exists a compact neighborhood K of x, the restriction F\k is upper semicontinuous on K, having a compact graph.