Can someone please show me one example of a continuous function: $f : X\subset \mathbb{R} \rightarrow \mathbb{R}$ but $\lim_{x\rightarrow a} f(x) \neq f(a)$ with $a \in X$?
The converse is clear for me, that means if $\lim_{x\rightarrow a} = f(a)$ then $f$ is continuous at $a$.
I was trying to look for some definition of $f$ where we'd have $a \in X' \setminus X$...
I'd appreciate some help! Thanks!
On
If $f: X=(0,1) \cup \{2\} \to \mathbb{R}$ is defined by $f(x)=\sin(\frac1x)$ for $x \in (0,1)$ and $f(2)=42$ then $f$ is continuous on $X$ (continuity is trivial on isolated points: take $\delta=1$ for any $\epsilon>0$, e.g.) but $\lim_{x \to 2} f(x)$ does not exist as $2 \notin X'$ (but $2 \in X$); and as a bonus $\lim_{x\to 0} f(x)$ does not exist even though we do have $0 \in X'$.
Or take any function defined on $X=\mathbb{Z}$. No limit exists to points of $X$ (or outside) as $X'=\emptyset$.
I think your confusing things here: it $\;f:X\to\Bbb R\;$ is continuous, then for any $\;a\in X\;$ it must be true that
$$\lim_{x\to a\\x\in X}f(x)=f(a)\;$$
This is just part of definition (or of what follows from it, depending on your particular definition of continuity).
The above has nothing to do with the fact that if $\;\{x_n\}\subset X\;$ , then $\;\lim\limits_{n\to\infty} x_n=x\in X\iff x\in X'\;$ , which is perhaps what you're thinking of.