PomeDF0FD081C3EFCorollary 16.28.39.DF0FD081C3EF For fixed $a$ and $\delta >0$, the map $q\mapsto F_{\delta ,q}(a)$ on $C(K,\mathbb R)$ is Gâteaux differentiable at $q$ if and only if all pairs active at $(\delta ,q)$ have the same probability measure $\mu $. In that case its derivative is
Proof. With $s=0$, (16.28.36) shows that the right derivative in the direction $r$ is the minimum of $\int _ Kr\, \mathrm{d}\mu $ over active pairs, and the direction $-r$ shows that the left derivative is the maximum. If all active pairs have the same probability measure, these values agree and define a continuous linear derivative. Conversely, if the map is Gâteaux differentiable, the minimum and the maximum agree for every continuous $r$. Continuous functions separate Borel probability measures on $K$, so all active pairs have the same probability measure; their constant is then determined by (16.28.32), since $\delta >0$. $\square$
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