# Possible error in given definition of uniform continuity of a function on a metric space. [duplicate]

I have the following defitions

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?

• 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. Oct 7, 2022 at 21:00
• 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. Oct 7, 2022 at 21:04
• 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? Oct 7, 2022 at 21:05
• Please add the relevant information so that it does not rely on the image alone. Oct 7, 2022 at 21:06