I have the following defitions

enter image description here

I don't really see a difference between the two definitions. Shouldn't the first one be with for all x and there exists \delta switched?

If there is no mistake, could someone please explain the difference?

  • $\begingroup$ Please do not use images to convey information not otherwise present in your post. See here for an explanation of why this is bad practice. $\endgroup$ Oct 7, 2022 at 21:00
  • $\begingroup$ The difference is that the first definition is "anchored" at $y$: it's about the behavior of $f$ near a fixed $y$. The second is for all $x,y$; it's about the behavior of $f$ at any pair of close-enough points. Consider $f(x)=1/x$ on $(0,1). It is continuous at each point, but not uniformly continuous because you can always find two points that are as close as you specify, but whose images are very far apart. $\endgroup$ Oct 7, 2022 at 21:04
  • $\begingroup$ I understand now thank you. I think that I missed the "is continuous at y" bit from the definition. Can you please post this as an answer so that I can mark it as correct? $\endgroup$ Oct 7, 2022 at 21:05
  • $\begingroup$ Please add the relevant information so that it does not rely on the image alone. $\endgroup$ Oct 7, 2022 at 21:06


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