Discontinuous robust mappings are approximatable
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- by Shu Zhong Shi, Quan Zheng and Deming Zhuang PDF
- Trans. Amer. Math. Soc. 347 (1995), 4943-4957 Request permission
Abstract:
The concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization. A set $A$ of a topological space $X$ is said to be robust iff ${\text {cl}}A = {\text {cl}}$ int $A$. A mapping $f:X \to Y$ is said to be robust iff for each open set ${U_Y}$ of $Y$, ${f^{ - 1}}({U_Y})$ is robust. We prove that if $X$ is a Baire space and $Y$ satisfies the second axiom of countability, then a mapping $f:X \to Y$ is robust iff it is approximatable in the sense that the set of points of continuity of $f$ is dense in $X$ and that for any other point $x \in X$, $(x,f(x))$ is the limit of $\{ ({x_\alpha },f({x_\alpha }))\}$, where for all $\alpha$, ${x_\alpha }$ is a continuous point of $f$. This result justifies the notion of robustness.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 4943-4957
- MSC: Primary 90C48; Secondary 49J45, 54C08
- DOI: https://doi.org/10.1090/S0002-9947-1995-1308024-X
- MathSciNet review: 1308024