By Vladimir F. Dem'yanov, Alexander M. Rubinov

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**Extra resources for Quasidifferential Calculus (Translations Series in Mathematics and Engineering)**

**Example text**

1'. 1 be satisfied and let, in addition, the mapping H be uniformly directionally differentiable. Then the function u(z) = f(H(z)) is uniformly directionally differentiable at the point x. P r o o f. 1 that u(x+ag) - u(x) - au'(x,g) = a[f'(H(x), HX (g) + y'8 (a)) - f'(H(x), HX (g))] + o(ava (g)) . Here va (g) = HX (g) + Vg (a). Since the mapping is uniformly differentiable, we have y'8 (a) a -- 0 0 uniformly with respect to g E B. Therefore, the quantity in square brackets tends to zero (as a -4 +0) uniformly with respect to g, the same being true for the quantity o(ava (g))/a.

2. If a function f is convex on a convex open set S c R", then f the following limit exists: df(x) a8 = 1 im 1 [f(x+ag) a a -f(x)] . 2. Let a function f be convex on Rn, x E Rn. 5) 0 is called the subdifferential of the function f at the point x. Any element v e q)I(x) is called a subgradient (or generalized gradient) of the function f at x. The mapping ) q : Rn - 17(Rn) is called the subdifferential mapping (related to the function P. 3. The set oY(x) is nonempty, convex, closed and bounded. 1.

11) (b) If h+ (a) > 0 for some a E [a,b], then h(,(3) > h(a) V fl > a. 12) Property (a) follows immediately from the lemma. , there exists a f3 > a such that h(f3) = h(a). 11) we conclude that h(f3') = h(a) V f3' E [a, f3]. 12). 2. 1 be satisfied and let inf h+ (a) ,r. m= aE [a, b] , M= sup h+ (a) . aE [a, b] Then m(b-a) <_ h(b) - h(a) <_ M(b-a) 0 P r o o f. 1 to the functions hl (a) = Ma - h(a) and . 1. Let a function f be defined and continuous on an open set X c Rn and differentiable in a direction g at every point of the interval [x,x+aog] _ {x+ag I a E [O,ao]}, where ao > 0 and x is fixed.