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The set 0, first constructed by Cantor, is not, however, enumerable. To prove this we observe that any number x in the interval 0 ~ x ~ l can be expressed in the form where each numerator an is either 0, l, or 2. The points of E 1 can be expressed as X= 58 Differentiation of monotone functions the points of E 2 as 2: ;~+l+(O or 00 x = j); n=3 the points of E 3 as 00 1 +(O or 32 or 9 2 or x -_ "' ~ an+ 3n 27 2+2) 9 . e. 2: ~k 00 x = and an = l. e. the points of the interval (0, l) that remain after the removal of Ev E 2 , ••• , can be represented only in the form 2: ;z, 00 x = k=l where each ak is either 0 or 2.

G is an For there is a neighbourhood of g in which o(x) = o(g) = I. e. g is an interior point of o(x). 3. If K(x) is a closed set, then it is identical with its closure. Let g1 , g2 , ••• be any convergent sequence of points belonging to g. 2. e. the set K(x) contains the limit of any convergent sequence of points in K(x). The preceding theorems show that the definitions of open and closed sets in terms of the continuity of their indicators are equivalent to the usual definitions. 3. Two sets a(x) and fJ(x) are said to be 'complementary' with respect to a set y(x) if a(x) and fJ(x) are disjoint, and y(x) is their union.

Let g1 , g2 , ••• be any convergent sequence of points belonging to g. 2. e. the set K(x) contains the limit of any convergent sequence of points in K(x). The preceding theorems show that the definitions of open and closed sets in terms of the continuity of their indicators are equivalent to the usual definitions. 3. Two sets a(x) and fJ(x) are said to be 'complementary' with respect to a set y(x) if a(x) and fJ(x) are disjoint, and y(x) is their union. 4. The necessary and sufficient condition that a(x) and fJ(x) should be complementary with respect to y(x) is that a(x)+fJ(x) = y(x).

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A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems by Vakhrameev S.A.


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