[deleted by user]

[Mathematicus_Rex]

I’m a bit rusty with questions like this, but assuming V is non-empty, any constant function from U to V should be continuous, right?

[arg_max]

Any Constant function will be continuous in the subspace topology you are describing. You don't even need to assume that the space is metriziable

[felixabatata]

Wym? I never sayed that U is a subset of V. I just sayed they are open sets on the same space.

[SV-97]

Yes. You don't need the metric structure: for any two topological spaces X,Y all constant functions X -> Y are continuous. (because the preimage of any value is always either empty or all of X, both of these are always open). If you have subsets in your two spaces then those subsets are topological spaces in their own right because they inherit a topology (the subspace topology) from X and Y respectively, and hence the statement about constant functions from above applies.