Proposition 16.28.35.FDDE0F20FBB2 For $r\in C(K,\mathbb R)$ and $s\in \mathbb R$, one has
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16.28.36
\begin{equation} \lim _{h\downarrow 0} \frac{F_{\delta +hs,q+hr}(a)-F_{\delta ,q}(a)}h =\min _{(\mu ,c)\in \mathcal J^{\mathrm{act}}_ a(\delta ,q)} \left(\int _ Kr\, \mathrm{d}\mu +sc\right). \end{equation}
The same limit holds along any sequence $h_ j\downarrow 0$ if $s,r$ in the difference quotient are replaced by $s_ j,r_ j$, where $s_ j\to s$ and $r_ j\to r$ uniformly. Thus the one-sided directional derivative is stable under convergence of the directions.
Proof.
By compactness of the set of active pairs, the minimum on the right is attained. For a fixed pair $(\mu ,c)$ active at $(\delta ,q)$, the representation (16.28.28) at the perturbed parameters gives
\[ F_{\delta +hs,q+hr}(a) \leq F_{\delta ,q}(a)+h\left(\int _ Kr\, \mathrm{d}\mu +sc\right). \]
Choosing an active pair that attains the minimum on the right, we see that the upper limit of the difference quotients is at most this minimum. Conversely, choose a pair $(\mu _ h,c_ h)$ active at $(\delta +hs,q+hr)$. The representation at $(\delta ,q)$ gives
\[ \frac{F_{\delta +hs,q+hr}(a)-F_{\delta ,q}(a)}h \geq \int _ Kr\, \mathrm{d}\mu _ h+sc_ h. \]
The estimate (16.28.34) bounds the difference quotients. Along a sequence of values of $h$ realizing their lower limit, lemma 16.28.33 gives a subsequence of pairs converging to a pair active at $(\delta ,q)$. For this pair, $\int _ Kr\, \mathrm{d}\mu +sc$ is at least the minimum on the right. This proves (16.28.36).
For convergent directions, the same two inequalities hold. The fixed active pair attaining the minimum gives an upper bound with the right limit. The active pairs at the perturbed parameters still have uniformly bounded constants, and their subsequential limits are active at $(\delta ,q)$. Uniform convergence of the $r_ j$ controls the integrals $\int _ Kr_ j\, \mathrm{d}\mu _{h_ j}$, and $s_ j\to s$ controls the terms $s_ jc_{h_ j}$. The lower bound therefore has the same limit. Note that all perturbed growth allowances $\delta +h_ js_ j$ are positive for small $h_ j$.
$\square$
Pome
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