I have two questions.
- PMA, Rudin p.97 proves the existence of a monotonic real function $f$ defined on $(a,b)$ such that $f$ is continuous on $E$- a countable subset of $(a,b)$ and discontinuous on $(a,b) \setminus E$.
Then, it states that "It should be noted that the discontinuities of a monotonic function need not be isolated".
I don't understand how come the existence of above function implies that. Isn't $f\upharpoonright ((a,b)\setminus E)$ continuous on its domain?
- PMA, Rudin p.98 states an equivalent definition of limit defined by \epsilon-\delta.
That is, when f:E\rightarrow X$ is a function from a metric space to another, $\lim_{t\to x} f(t) = A$ iff [For every neighborhood $U$ of $A$, there exists a neighborhood $V$ of $x$ such that $V\bigcap E≠\emptyset$ and $t\in (V\bigcap E)\setminus \{x\} \Rightarrow f(t)\in U.
This is exactly the same as the definition by \epsilon-\delta$. However, this should be equivalent to $\lim_{t\to x} f(t) = A$ only if $x$ is a limit point of $E$. Does the definition above has an information that $x$ is a limit point of $E$?