Non differentiability at the energy wells

Let $X\subset[0,1]\cap\mathbb Q$ be such that

$$\Big(\inf_{\theta\in X}\{\widehat Q_{\bf m}f(\theta,z)\}\Big)^{\ast\ast}=\widehat Q_{\bf m}f(z)$$

for all $z$, and let $\theta^\ast\in X$ be an energy well satisfying

$$\Big(\inf_{\theta\in X\setminus\{\theta^\ast\}}\{\widehat Q_{\bf m}f(\theta,z)\}\Big)^{\ast\ast}>\widehat Q_{\bf m}f(z)$$

for some $z$. Non differentiability at the energy wells. The function $\theta\mapsto Q_{\bf m}f(\theta,z)$ is not differentiable in $z$ at $\theta^\ast$. This concerns the regularity of the reduced energy with respect to the parameter $\theta$ at essential energy wells, which are locking states; the source gives no resolution of this conjectural non-differentiability claim.