By Richard B. Holmes (auth.)

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Under the hypotheses is a continuous With (i) the two-sided lira t+0 on X. the same hypotheses, -f'(Xo;-X ) = min Hence function of the Theorem, directional {~(x): ~ E ~f(Xo)}. derivative f(Xo+tX) -f(x o) t is show that 28 exists and has the value k constant on the set (with value X) if and only if the function ~f(x o) . x g X We note further that the set of (I) exists is a closed subspace of fines a continuous and only if Proof. X linear functional c) Corollary. for which the limit in on which the value of (i) de[26, p.

It is also easy to see that these are equivalent to the condition X × R I''. the set of all isc proper convex We denote by F(X) "f "epi (f) is closed in 46 functions on X. Theorem. f** Proof. hence If made theorem (i) ~ x o e dom (Xo,f**(Xo)) t o c R1 % epi such closed in a). f set Now t o < 0, assume when (f) C (f**) (f), dom such {tot+ < X , Y o > for o t h e r w i s e since otherwise t o = -i. Then t = f(x). that a contradiction convex such g Then (dom (f)). Then < f(Xo). Yo E X* Hence set. and > : (x,t) the sup in for given to Young's With e epi (f)}.

Ko < 0 This implies for which Hence • - of -- and e i=O follows that by (3), so Halkin of Ji' ~ 0 where sup Ko,Y ° Yo ~ K°o J is a by 15d). + an = I, Thus ~ Xi = 0. ,n; x ~ K, -< - Z 1(~i + (x,Yi~) of this theorem have been given by Vlach from Ioffe-Tikhomirov so in it -< 0 also, qed. [23], and Pshenichnii variational ), Yi ~ -Xoai J° ~ Xi JO" From this, and the fact that 1 be seen shortly, and Since = co (J U . . n, loss of and let Jo - sup ~ K i , Y i / ~ i ~ since Yi ~ K~. , such that [31].