I think there is a subquestion, or related (!) question, namely why is there little functional analysis of partial functions? Teaching real analysis to students makes it clear that it is all about partial functions $\mathbb R \to \mathbb R$ , each of which has a domain and range, which we often want to know. Now the solutions of differential equations with a parameter are often regarded as smooth in the parameter, but usually only with fixed domain. For example it is reasonable to suggest that the family of partial functions $f_y: x \mapsto \log(x+y)$ varies continuously with $y$, but of course the domain varies with $y$, so the answer is not so clear, especially if you suggest, why not?, that $f_y$ is, or should be, a smooth function of $y$.
Actually a web search on "partial functions" gives quite lot of hits, but I am not sure of anything definitive.