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I am having trouble understanding exactly the difference between the epsilon delta definition for continuity and the one for the limit of a function.

  • epsilon greater than 0, there exists a δ such that |x|<δ implies |f(x) - f(0)|< epsilon

  • epsilon greater than 0, there exists a δ such that 0 <|x|<δ implies |f(x) - f(0)|< epsilon

From R--> R if f is a continuous function which one must be true? I feel like they are they are both saying the same thing and I think both are true. I know the first is true because that is the definition of continuity, but I don't quite get the difference in the 2nd. the greater than 0 part confuses me. Are they both equivalent definitions or not?

I also have some confusion about what it means to be continuous in terms of images and preimages.

a. I know that if a function is continuous, then it's preimage of any open intervel in S in R, is open in R. I think this MUST be true(?) b. The image f(S) of any open interval S in R is open in R. I believeThis does NOT necessarily have to be true to about the function for it to be continuous

These aren't really homework questions, but things I have in my notes and were told to think about in preparation for the next class.

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No need for the edit, we can tell this isn't homework :) –  wj32 Oct 31 '12 at 22:50
    
OH ok phew, I always get paranoid about it seeming like hw when I write so much haha –  george wyatt Oct 31 '12 at 22:52
    
You're totally right, a "limit point" is definitely not what I meant. I should edit the post. –  george wyatt Oct 31 '12 at 23:08
    
Also, please consider accepting the answers to your previous questions by clicking the green tick. –  wj32 Oct 31 '12 at 23:20

1 Answer 1

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Consider this function: $$f(x)=\frac{\sin x}{x}.$$

It is not defined at $0$, so we cannot talk about whether $f$ is continuous at $0$. Yet $\lim_{x \to 0} f(x)$ exists and is equal to 1. This is the essential difference between continuity at a point and the limit at a point. For the latter, we explicitly ignore the value of the function at the point, because we do not care. (This is where the "$0<|x-\cdots$" in the definition of the limit comes from.) Note that if $a$ is a limit point of $f$, then $f$ is continuous at $a$ if and only if $$\lim_{x \to a} f(x) = f(a).$$

Another difference is that $\lim_{x \to a} f(x)$ only makes sense if $a$ is a limit point of the domain of $f$, whereas $f$ will automatically be continuous at $a$ if $a$ is an isolated point. For example, consider a function that is only defined at the integers.

Now for your other questions:

a. i), ii) Yes, this is correct. See this question.

b. i), ii) Again, correct. Think about minimums and maximums for easy counterexamples.

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Thank you. This really cleared things up for me. –  george wyatt Nov 1 '12 at 0:32

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