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I was thinking about the notation of function mappings, such as $$f: \mathbb R^n \rightarrow \mathbb R^m,$$ and wondered if there would be any point to considering functions with variable output (or input) dimensions. In this case I would imagine we could define \begin{gather} f:\mathbb R^n\rightarrow \mathbb R^{g(x)}, \\ g:\mathbb R^n\rightarrow\mathbb R^1, \end{gather} or something like that (I don't like the $x$ in the definition). I could even provide a simple example of such a function: $$ f(x)=\begin{cases} x & x<0, \\ (x,x^2) & x\ge0. \end{cases} $$ My question is whether this concept already exists, and if so what purpose does it serve? I can vaguely think of it being useful in bifurcation theory, where for some values of a parameter a quantity may have a single instance and for other values it may have multiple.

Michael M
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    A function is a map between two sets so you would have to define a codomain that includes objects of differing dimensions. To handle your case, we would normally define $f(x) = (x,0)$ for $x\lt 0$. – John Douma Mar 04 '24 at 21:12
  • One definition you might find interesting (as the codomain does depend on a function evaluation) is that of the differential:

    Given smooth manifolds $M,N$ and a smooth map $\phi: M\rightarrow N$, for any $x\in M$ we usually define the differential of $\phi$ at $x$ as a map $$d\phi_x: T_x M \rightarrow T_{\phi(x)}(N)$$

    So we see dependence of the codomain on the function $\phi$ (but not on $d\phi_x$ itself!).

     – Theo C. Mar 04 '24 at 22:03

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