Michael selection theorem

In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:
Examples.
A function that satisfies all requirements.
The function: formula_3, shown by the grey area in the figure at the right, is a multi-valued function from the real interval [0,1] to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: formula_4 or formula_5.
A function that does not satisfy lower hemicontinuity.
The function
formula_6
is a multi-valued function from the real interval [0,1] to itself. It has nonempty convex closed values. However, it is not lower hemicontinuous at 0.5. Indeed, Michael's theorem does not apply and the function does not have a continuous selection: any selection at 0.5 is necessarily discontinuous.
Applications.
Michael selection theorem can be applied to show that the differential inclusion
has a "C"1 solution when "F" is lower semi-continuous and "F"("t", "x") is a nonempty closed and convex set for all ("t", "x"). When "F" is single valued, this is the classic Peano existence theorem.
Generalizations.
A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to an equivalence relating approximate selections to almost lower hemicontinuity, where formula_8 is said to be almost lower hemicontinuous if at each formula_9, all neighborhoods formula_10 of formula_11 there exists a neighborhood formula_12 of formula_13 such that formula_14 
Precisely, Deutsch–Kenderov theorem states that if formula_15 is paracompact, formula_16 a normed vector space and formula_17 is nonempty convex for each formula_9, then formula_8 is almost lower hemicontinuous if and only if formula_8 has continuous approximate selections, that is, for each neighborhood formula_10 of formula_11 in formula_16 there is a continuous function formula_24 such that for each formula_9, formula_26.
In a note Xu proved that Deutsch–Kenderov theorem is also valid if formula_16 is a locally convex topological vector space.