Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 0 down vote accepted

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.

share|cite|improve this answer
Thank you. This really cleared things up for me. – george wyatt Nov 1 '12 at 0:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.