There are a few natural ways to construct a product metric space from pre-existing ones (article) but what condition do we need on spaces/ functions for it to be that the continuity of component function (w.r.t metric components) imply continuity of whole function on the product space?
This post seems to answer it is not easy but this other post does it for the 2-norm on $\mathbb{R}^d$ . I feel the result in the second mentioned post could be at least generalized for different metrics on $\mathbb{R}^d$, could it? What would be the generalized result?
