Let $L>0$ and $\Omega \subset \mathbb{R}^n$ a bounded Lipschitz domain. Define $$ B_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \forall s,t \in [0,1]\}. $$ I would like to show that, for every fixed $s,t \in [0,1]$, the functional $f \mapsto \int_\Omega |f(t,x)-f(s,x)|$ is continuous with respect to $L^2((0,1) \times \Omega)$-norm on the set $B_{\frac12,L}$, . It seems to me that this amounts to the question of continuity of point evaluation on a set of Hölder continuous functions with respect to $L^2$-norm, but I was not able to show it.

Does anyone have any direction: reference, counterexample, proof (hopefully)?