I have a parameterized optimization problem \begin{eqnarray} \max_{x\in D(\theta)} f(x,\theta). \end{eqnarray} Assumptions of the standard Berge's maximum theorem are satisfied, so the value function $f^*(\theta)$ is continuous on my state space $(0,1)$.
That $f^*(\theta)$ being continuous on $(0,1)$ doesn't exclude some erratic behavior that are unlikely to happen in my application scenario. Is there any extra conditions that could impose more regularity on the value function, e.g. not just continuous but also lipschitz? (I want $\lim_{\theta\to 0 \mbox{ or }1} f^*(\theta)$ exists.) Unfortunately, I cannot impose standard assumptions like quasiconcavity or supermodularity.
I have done my literature search. There are many generalizations aiming at weakening the condition of maximum theorem, but not so many aiming at strengthen the result.
Many thanks for any comments or references.
