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I believe that to prove a function f is well defined you need to prove that for any two inputs a and b, a = b → f(a) = f(b). But what if we have a function where it makes sense to have different outputs assigned to the same input? For example, an ellipse in R^2, or a hyperbola symmetric about the y axis in R^2. By the standard of before the functions for such objects are not well defined. Is this something we just accept, because they are otherwise “nice”? If yes what exactly would that mean, and why do we overlook ill-definedness in this case but not others? Or if no, is there a different standard for evaluating such objects as well defined?

Joa
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  • Q: "But what if we have a function where it makes sense to have different outputs assigned to the same input?" A: Look instead at a function that maps to $\mathbb R^n,.$ – Kurt G. Sep 05 '23 at 03:53

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There are two notions here: a function and a curve. Ellipses, hyperbolae etc. are curves.

Given a function $f:D\to \mathbb{R}$ there is a curve called the graph of $f$ defined by $\Gamma_f=\{(x,f(x))|x\in D\}$, where $D\subset \mathbb{R}$ is the domain of $f$. So, for instance, the function $f:x\mapsto x^2$ has a graph which is a curve called a parabola.

To every function we can associate a curve in this way, but not every curve must come from a function, and your examples show why not. So, a curve is more general.

Joshua Tilley
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