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In Hubbard's Vector Calculus, I saw an unconventional definition for the limit of a function:

Definition: Let $X$ be a subset of $\mathbb{R}^n$. A function $\mathbf{f}:X\rightarrow\mathbb{R}^m$ has the limit $\mathbf{a}$ at $\mathbf{x_0}$:

$$ \lim_{\mathbf{x} \to \mathbf{x_0}} \mathbf{f(x)} = \mathbf{a} $$ if $\mathbf{x_0}$ is in the closure of $X$, and for every $\epsilon>0$, there exists $\delta>0$ such that for all $\mathbf{x}\in X$, $|\mathbf{x} - \mathbf{x_0}| < \delta$ implies $|\mathbf{f(x)} - \mathbf{a}| < \epsilon$.

The peculiarity of the definition is the usage of "$|\mathbf{x} - \mathbf{x_0}| < \delta$" instead of $0<|\mathbf{x} - \mathbf{x_0}| < \delta$.

I think this subtle difference can have rather significant implications. Using this definition, if $ \lim_{\mathbf{x} \to \mathbf{x_0}} \mathbf{f(x)}$ exists, then $\mathbf{f(x)}$ is continuous at $\mathbf{x_0}$, right? Are there other profound implications you can think of?

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    $\begingroup$ If the limit exists and $\mathbf{x}_0$ is in the domain of $\mathbf{f}$. It's not much of a difference, you take the limit of the restriction to the complement of $\{\mathbf{x}_0\}$ if you need the other concept of limit. That definition isn't actually so unconventional. In French and German texts at least, it is frequently used [I haven't read enough books to say it is the standard definition there, but it may be]. $\endgroup$ – Daniel Fischer Jun 11 '15 at 7:15
  • $\begingroup$ @DanielFischer: just wanted to clarify what you said. #1. If the limit exists and $\mathbf{x_0}$ is in the domain of $\mathbf{f}$, then $\mathbf{f(x)}$ is continuous at $\mathbf{x_0}$, right? #2. You say "the limit of the restriction to the complement of {$\mathbf{x_0}$}". I'm rather confused by this - is this a limit of some sequence of sets? Thanks. $\endgroup$ – FreshAir Jun 11 '15 at 17:16
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    $\begingroup$ If $\mathbf{x}_0$ is in the domain of $\mathbf{f}$, then $\mathbf{f}$ is continuous at $\mathbf{x}_0$ if and only if the limit of $\mathbf{f}$ at $\mathbf{x}_0$ exists. And if we let $\mathbf{g} = \mathbf{f}\lvert_{X\setminus \{\mathbf{x}_0\}}$, then $\lim\limits_{\mathbf{x}\to\mathbf{x}_0} \mathbf{g}(\mathbf{x})$ is the limit of $\mathbf{f}$ at $\mathbf{x}_0$ in the $0 < \lvert\dotsc\rvert < \delta$ definition. That is often denoted $\lim\limits_{\substack{\mathbf{x}\to\mathbf{x}_0\\ \mathbf{x}\neq \mathbf{x}_0}} \mathbf{f}(\mathbf{x})$. $\endgroup$ – Daniel Fischer Jun 11 '15 at 19:45
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No, there aren't really, since you hit the nail on the head. This alternate definition of a limit is the standard definition of continuity. So any other implications of this alternate definition will stem from the fact that limits have been redefined as continuity.

You could say that one implication is that the intuitive meaning of a limit no longer holds. For example, define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = 0$ for $x = 0$ and $f(x) = 1$ for all $x \ne 0$. The alternate definition would say that there is no limit at $x = 0$. (Normally, you would say that $f$ is not continuous at $x = 0$.

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