# Limit of function at non limit point

Given metric spaces $$(X,d_X), (Y, d_Y),$$ where $$S\subseteq X,f:S\to Y,$$ and $$x_0$$ a limit point of $$S$$, we say $$y_0$$ is a limit of $$f$$ at $$x_0$$ when $$\forall \epsilon>0, \exists \delta>0 \text{ s.t. } f(B_\delta(x_0)\cap S\setminus \{x_0\})\subseteq B_\epsilon(y_0).$$

To be sure, since $$f(\emptyset)=\emptyset \subseteq B_\epsilon(y_0)\quad \forall y_0\in Y,$$

can we say any point $$y_0\in Y$$ is a limit of $$f$$ at $$x_0$$ when $$x_0$$ is not a limit point of $$S$$? Or does the definition of a limit of a function not apply to non limit points in the first place?

If $$x_0\in X$$ is not a limit point of $$S$$ , then $$\exists\delta>0$$ such that $$B_{\delta}(x_0) \cap S\setminus \{x_0\}=\emptyset$$
Then $$f(\emptyset) \subset B_{\epsilon}(y_0)$$ for all $$\epsilon>0$$
Hence if $$x_0\in X$$ is not a limit point of $$S$$ then every point $$y_0\in Y$$ is a limit point of $$f$$ at $$x_0$$.
• Right I got that; I guess my question was whether the "limit of a function" formally applies to non limit points; the notes I have assume $x_0$ is a limit point in the definition. Commented Aug 15, 2022 at 8:04
• According to my answer if $x_0$ is a non limit point then every $y\in Y$ is a limit point of $f$ at $x_0$. So the definition is vacuously satisfied and "non-limit points" are useless here. Commented Aug 15, 2022 at 8:25