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This has been shown by Dudley, see https://www.jstor.org/stable/24490947.

.$$ In view of convexity of $f$ and $S$ this last condition implies (actually it is equivalent to) global optimality of $x$. …

Then, the first order (necessary and - due to convexity - sufficient) optimality conditions of your first problem read $$f'(x_0;d) \ge 0 \quad\forall d \in T_C(x_0),$$ where $T_C(x_0)$ is the tangent cone …

Yes, the converse is also true. This follows from the answer in https://math.stackexchange.com/questions/4227159/characterization-of-lipschitz-derivative. In fact, your condition yields $$ | (\nabla f …

You can take $X = \ell^\infty$ (actually $\ell^p$ with $p > 2$ should work) and define $$ f(x) = \frac12 \| x \|_{\ell^2}^2. $$ Note that $f$ equals $\infty$ on $\ell^\infty \setminus \ell^2$. One can …