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.