Corollary 16.28.240D01C895AF55

Proof. At $\tau $ one has $F_\tau (a)=\psi (a)$, and raw contact holds for all $\delta \geq \tau $ by theorem 16.28.14. If $D_\tau (a)=0$, then by (16.28.21) every difference quotient at $\tau $ vanishes, so $F_\delta (a)=\psi (a)$ for all $\delta >\tau $. If $D_\tau (a)>0$, then $F_\delta (a)>\psi (a)$ for all $\delta $ slightly larger than $\tau $, and by monotonicity for all $\delta >\tau $. In this case $\Lambda _{\mathrm{reg}}(a)$ contains $\tau $ but no larger parameter, so by theorem 16.28.14 it equals $\{ \lambda \} $, where $\lambda =\tau $ is the smallest element of $\Lambda $. $\square$