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What's the difference between two concepts, 1. limit of function and 2. continuous function? I looked at their definitions in the textbook written by Rudin and it's so hard to differentiate between them.

They look similar to me.

[Edited] I can't find the difference between two definition. If they are the same, it means that a function is continous if and only if every point is a limit point. But this is not true. So I want to know what makes a difference between a point being a limit of function and function being itself continuous

Images of Definitions

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user86261
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  • @CameronBuie I have no idea why this is a duplicate of that question. I just read the question: Definition of continuity at a point, and it was asking about dropping the condition 0<… in dX(x,p)<δ. – user86261 Aug 07 '13 at 03:28
  • The other question asked about the difference between Rudin's definitions of function limit and continuity at a point, too. Were you curious about something else, instead? If so, please elaborate in your post, and we'll be glad to answer. – Cameron Buie Aug 07 '13 at 03:31
  • @CameronBuie My question is that I can't find the difference between two definition. If they are the same, it means that a function is continous if and only if every point is a limit point. But this is not true. So I want to know what makes a difference between a point being a limit of function and function being itself continuous – user86261 Aug 07 '13 at 03:41
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    Could you write the two definitions? I don't have the Rudin's book at hand. – Makoto Kato Aug 07 '13 at 03:52
  • This question would seem to be related, then, even though it isn't about Rudin's definitions, specifically. I second Makoto's request to add the definitions in question to your post. Also, you should explain in your post (since not everyone reads the comments) what is troubling you about the definitions. – Cameron Buie Aug 07 '13 at 03:56
  • @MakotoKato I've attached the image of two definitions! – user86261 Aug 07 '13 at 05:02
  • Thanks. However, I'm sorry that nobody can answer your question anymore because it was marked as duplicate. – Makoto Kato Aug 07 '13 at 05:21
  • Now that you've posted your definitions, it's even clearer that your post is indeed a duplicate of the other. The other user simply noticed the only difference between the definitions of function limit and continuity at a point. (Note the difference between continuity at a point and being a continuous function. Also, did you compare with the remark after definition 4.1?) Rudin's definition does not imply that every point of the domain is a limit point, as he proceeds to explain after Definition 4.5 ("It should be noted....") – Cameron Buie Aug 07 '13 at 05:38
  • @Makoto: You have enough reputation that you can always vote to reopen. (Also, you can request reopening at the appropriate meta page, if you like.) – Cameron Buie Aug 07 '13 at 05:43
  • While the definition $4.1$ does not require that $f(x)$ is defined at $p$, the definition $4.5$ requires that $f(x)$ is defined at $p$. Moreover the definition $4.5$ requires that $lim_{x \rightarrow p} f(x) = f(p)$. – Makoto Kato Aug 07 '13 at 07:56

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The limit (if it exists) is the number the function approaches. In precise terms $L$ is the limit of $f$ at $a$ if for every $\epsilon > 0$, there is a $\delta > 0$, so that for every $x$, $0 < |x - a| < \delta \implies |f(x) - L| < \epsilon$. When the function is continuous at $a$, the number it approaches at $a$ is $f(a)$. So in the definition above, $L$ is replaced by $f(a)$. You can also change the first part to $|x - a| < \delta$ since the statement is clearly true for $x = a$ also. That's why the definitions look so similar.

In the book, notice that for limits, it uses a general variable $q \in Y$ but for continuity, it is replaced with $f(p)$.

Pratyush Sarkar
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