# Implications of Alternate Definition of the Limit of a Function

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?

• 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]. – Daniel Fischer Jun 11 '15 at 7:15
• @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. – FreshAir Jun 11 '15 at 17:16
• 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 $\lim\limits_{\substack{\mathbf{x}\to\mathbf{x}_0\\ \mathbf{x}\neq \mathbf{x}_0}} \mathbf{f}(\mathbf{x})$. – Daniel Fischer Jun 11 '15 at 19:45

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 $x = 0$.