Introductory real analysis defines continuity via limits. For instance, the function $f:\mathbb{R}\to\mathbb{R}$ is continuous at $a \in\mathbb{R}$ iff $\lim\limits_{x\to a}{f(x)=f(a)}$. The definition of the limit, in turn, assumes $a$ to be an $(\mathbb{R},|\cdot|)$-accumulation point of the domain of $f$ (here the domain was $\mathbb{R}$ itself).

For general metric spaces, the criterium of being an accumulation point commonly seems to be dropped from the definition of continuity.

Definition (loosely put). Let $(X, \varrho)$ and $(Y, d)$ be non-trivial metric spaces. Let $f:X\to Y$, and $x_0\in X$. We say $f$ is continuous at $x_0$ iff $$\forall\varepsilon>0\ \exists\delta>0\ \forall x\in X\quad \varrho(x, x_0)<\delta\Longrightarrow d\left(f(x), f(x_0)\right) < \varepsilon.$$

So here is my admittedly subjective question.

Q: Why do we presuppose a point is an accumulation point of the domain in real analysis, and drop it for general metric spaces? What is the payoff for these asymmetric definitions?

Or are the definitions of continuity for general metric spaces (displayed above) and real analysis less ubiquitous than I think? Are there well-known authors who keep the criterium of being an accumulation point for points where we define continuity for general metric spaces?

Note. I understand that the restriction to accumulation points allows one the soundness of mind to ignore cases where $|x - x_0| < \delta$ is trivially true. On the other hand, removing the criterium of being an accumulation point expands the cases where one can apply the definition of continuity. But these reasons do not really differentiate between $(\mathbb{R},|\cdot|)$ and more general metric spaces. So the question is, why do we prefer to differentiate the definition of continuity at a point for real analysis and metric spaces.

See also: first comment under question.

  • $\begingroup$ From OP: I'm OK with this being made a community wiki. The remark loosely put means I didn't bother to define the function wrt to subsets of the metric spaces at hand. Also note that when a limit of a function at a point is defined for metric spaces, one usually does retain the condition of an accumulation point, and in this case, $\lim\limits_{x\to x_0}{f(x)=f(x_0)}$ simply may not hold when $f$ is continuous at $x_0$. $\endgroup$ – Linear Christmas Apr 26 at 21:16

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