# 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.
Please read Kavi Rama Murthy's comment below about the 1st question.   • I am a native English speaker, and I agree Rudin should have said for – J. W. Tanner Feb 5 at 3:49
• @J.W.Tanner Thank you very much for your answer. – tchappy ha Feb 5 at 3:52
• 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
• 1. Yes "for" should be incuded. 2. Your vsrsion is mathematically accurate. 3. Offer your services to proof read that text. – William Elliot Feb 5 at 5:47
• "Proof-read" (often written "proof read") means to read something for the purpose of detecting and correcting linguistic, typographical, and punctuation errors, including the detection of omissions. It was suggested that you could or should offer your services to the publisher in order to produce an improved version of the text, as you seem to be capable of it. – DanielWainfleet Feb 5 at 13:20