TITLE: existence = well-defined?
QUESTION [5 upvotes]: When something (like a limit) is said to "exist" is this perfectly equivalent to "is well-defined"?  And, is "well-defined" more-or-less equivalent to, "computers could use this definition and there would (theoretically) be no miscommunications"?

REPLY [4 votes]: Well-defined has slightly different meanings in different mathematical context, but the idea is generally that it's defined independent from a particular representation.
For example, the following function $f\,:\, \mathbb{R} \to \{0,1\}$  $$
  f(x) = \begin{cases}
   1 &\text{if the first fractional digit is 9} \\
   0 &\text{otherwise}
  \end{cases}
$$
is not well-defined. Since we may write $1$ as either $1.000\ldots$ or $0.999\ldots$, the value the function takes depends on which particular representation of $1$ we pick.
There is a connection between existence and well-definedness, but the two aren't quite the same thing. For one thing, well-definedness is a stronger property. Saying "Let $x$ be the real number with $x^2 = 1$" doesn't well-define $x$, because it doesn tell us whether $x=1$ or $x=-1$. Existence is, however, necessary for well-definedness. If we attempt to define a function by placing impossible requirements on it - e.g. "Let $g$ be a real function such that $(g(x))^2 = x$ for all $x$", then $g$ is certainly not well-defined.