I'm seeking how to prove that Moreau-Yosida regularisation provides a continuous function: Given a convex function $f:X \rightarrow \mathbb{R}$ where $X$ is a compact subspace of an Euclidean space (namely, $\mathbb{R}^n)$, Moreau-Yosida regularisation, $g_{\eta}: X \rightarrow \mathbb{R}$ is a continuous function for any $\eta > 0$ where $g_{\eta} (x) = \inf_{x' \in X} \left( \frac{1}{2 \eta} \lVert x - x' \rVert^2 + f(x') \right )$ for all $x \in X$.
It is well-known that the gradient of $g_{\eta}$ is a Lipschitz continuous (so obviously $g_{\eta}$ is continuous) but I don't know how to prove the Lipschitzness.
The main question is whether $g_{\eta}$ is a continuous function for any fixed $\eta > 0$ or not. Can anybody help me?
