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Existence of quasicontinuous selections for the space $$2^{\mathbb R}$$. (English) Zbl 0863.54014
Let $$X$$ be an arbitrary topological space and $$F:X\to \mathbb R$$ a continuous multifunction with closed values. Then $$F$$ has a quasicontinuous and upper semicontinuous selection such that its discontinuity points form a nowhere dense set. Given an element $$(x,y)$$ of the graph of $$F$$ there is a quasicontinuous and upper semicontinuous selection $$g:X\to \mathbb R$$ of $$F$$ such that $$g(x)=y$$, $$g$$ is continuous at $$x$$ and its discontinuity points form a nowhere dense set.

##### MSC:
 54C65 Selections in general topology 54C08 Weak and generalized continuity
##### Keywords:
continuous multifunction; selection; quasicontinuity
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