# Question about continuous function in terms of limits of sequences

I am reading about continuous function, in this site http://en.wikipedia.org/wiki/Continuous_function specifically the section "Definition in terms of limits of sequences". My question is, let be $c\in \mathbb{R}$ an arbitrary element belonging to domain of $f$, where $f$ is in a closet and bounded set. Are there always a sequence $x_n$ which converges to $c$?.

-

1
Your question doesn't make much sense. You were asking about continuity and then you switched to closed sets. If $f$ a function or a subset? – Asaf Karagila Jul 19 '12 at 22:30

1
What about $x_n = c$ for all $n$? – Dylan Moreland Jul 19 '12 at 22:42

2
Why is $f$ in the closet? I thought we as a society were past that... – Quinn Culver Jul 19 '12 at 23:24

## 1 Answer

There is always the constant sequence $x_n = c$ for all $n$. In the extreme case that the domain of the function is $\{c\}$, there isn't anything else.

In general there is a dichotomy to be made according to whether $c$ is an isolated point of the domain $X$ of $f$. (Let's assume that $X$ is a subset of the real numbers $\mathbb{R}$, which seems to be the context of the question. In fact, with only mild notational change, this answer works in the context of arbitrary metric spaces.) We say that a point $c \in X$ is isolated if there is a $\delta > 0$ such that if $y \in X$ and $|x-y| < \delta$, then $y= c$.

For instance, if $f(x) = \sqrt{x^2(x-1)}$, the natural domain is $\{0\} \cup [1,\infty)$ and $0$ is an isolated point.

Now, for $X \subset \mathbb{R}$ and $c \in X$, the following are equivalent:

(i) $c$ is an isolated point of $X$.
(ii) Every sequence $\{x_n\}$ in $X$ which converges to $c$ is eventually constant: that is, $x_n = c$ for all sufficiently large $n$.
(iii) Every function $f: X \rightarrow \mathbb{R}$ is continuous at $c$.

In other words, an isolated point is something of a trivial case.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 36, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9516674876213074, "perplexity_flag": "head"}