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Corollary 16.28.377D611867F702

Corollary 16.28.37.7D611867F702 For fixed $a$ and $q$, the function $\delta \mapsto F_{\delta ,q}(a)$ has finite left and right derivatives $D^-_{\delta ,q}(a)$ and $D^+_{\delta ,q}(a)$ at every positive parameter, with

63336B576658 16.28.38
\begin{equation} D^-_{\delta ,q}(a)=\max _{\mathcal J^{\mathrm{act}}_ a(\delta ,q)}c, \qquad D^+_{\delta ,q}(a)=\min _{\mathcal J^{\mathrm{act}}_ a(\delta ,q)}c. \end{equation}

It is differentiable exactly when all active pairs have the same constant $c$. For each fixed $a,q$, the set of positive parameters where it fails to be differentiable is at most countable.

Proof. Apply (16.28.36) with $r=0$ and $s=\pm 1$. The direction $s=-1$ gives minus the left derivative, which proves (16.28.38). Concavity gives

\[ D^+_{\alpha ,q}(a)\geq D^-_{\beta ,q}(a) \qquad (0<\alpha <\beta ). \]

Hence the open intervals $(D^+_{\delta ,q}(a),D^-_{\delta ,q}(a))$, which are nonempty exactly at the parameters where differentiability fails, are pairwise disjoint. Choosing a rational number in each of them proves countability. $\square$

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