# Definition 4.25 on p.94 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

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.
• In my copy of Rudin there is no error. He says at any point $x$ instead of for any point $x$ and that is fine. @tchappyha – Kavi Rama Murthy Feb 5 at 5:41