I'm considering a generalised form of Moreau-Yosida regularisation.
Given a continuous function $f:X \times U \rightarrow \mathbb{R}$ where $f(x, \cdot)$ is convex for any $x \in X$ and $X$, $U$ compact subspaces of Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, resp., consider this parameterised function $f_{\eta}: X \times U \rightarrow \mathbb{R}$ defined as $f_{\eta}(x, u) = \inf_{v \in U} \left(\frac{1}{2 \eta} \lVert u - v \rVert^2 + f(x, v)\right)$ for any $\eta > 0$.
I wonder if $f_{\eta}$ is also a continuous function for given $\eta > 0$. In the previous question, the continuity of standard Moreau-Yosida regularisation can be shown by that the infimum of Lipschitz continuous functions is also Lipschitz continuous (A fortiori, it is continuous). However, in this case, I have no idea how to show the continuity with respect to the arguments $(x, u)$.
Can anybody help me? :)