I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.
Definition 4.25
Let $f$ be defined on $(a, b)$. Consider any point $x$ such that $a \leq x < b$. We write $$f(x+) = q$$ if $f(t_n) \to q$ as $n \to \infty$, for all sequences $\{t_n\}$ in $(x, b)$ such that $t_n \to x$. To obtain the definition of $f(x-)$, for $a < x \leq b$, we restrict ourselves to sequences $\{t_n\}$ in $(a, x)$.
It is clear that any point $x$ of $(a, b)$, $\lim_{t \to x} f(t)$ exists if and only if $$f(x+) = f(x-) = \lim_{t \to x} f(t).$$
(1) My 1st question is here:
I am very poor at English.
Is the following sentence correct as English sentence?
It is clear that any point $x$ of $(a, b)$, $\lim_{t \to x} f(t)$ exists if and only if $$f(x+) = f(x-) = \lim_{t \to x} f(t).$$
I guess the following sentence is correct:
It is clear that for any point $x$ of $(a, b)$, $\lim_{t \to x} f(t)$ exists if and only if $$f(x+) = f(x-) = \lim_{t \to x} f(t).$$
(2) My 2nd question is here:
Rudin didn't write as follows.
Why?
For any point $x$ of $(a, b)$, the following two statements are equivalent.
(a) $\lim_{t \to x} f(t)$ exists.
(b) $f(x+)$ and $f(x-)$ exist and $f(x+) = f(x-)$.
If (a) or (b) holds, then $$f(x+) = f(x-) = \lim_{t \to x} f(t).$$
By the way, my copy of "Principles of Mathematical Analysis 3rd Edition" is a paperback book printed in Malaysia. And I didn't intend to blame Rudin in my 1st question above. My English is very bad, so I wanted to know if the sentence is correct or not.
Please read Kavi Rama Murthy's comment below about the 1st question.
