Adherent point and limits I am reading Terence Tao Analysis I. I am confused in his remark 9.3.11. Can someone clarify this comment. If someone could also give me an example of a set that doesn't have any adherent point (This is probablyI don't understand remark 9.3.11 given in the image where the confusion arises)?
You are misunderstanding Tao. What he is saying is that he will not defined $\lim_{x\to x_0}f(x)$ when $x_0$ is not an adherent point of $X$. For instance, if $X=[0,1]$, he does not define the meaning of $\lim_{x\to2}f(x)$ (note that $2$ is not an adherent point of $[0,1]$).
An adherent point to a set is either an accumulation point or an isolate point.
If $ a$ is isolate , then we cannot define $\lim_{x\to a}f(x)$ .
For example, if $ D_f=E=[0,1]\cup \{2\}$, then $ 2$ is adherent but not an accumulation. So, it is not intetesting to speak about $\lim_{x\to 2}f(x)$.
Each element of a set is adherent to this set. Every non empty set contains adherent points.