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?
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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. – Arturo Magidin Oct 07 '22 at 21:00
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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. – Arturo Magidin Oct 07 '22 at 21:04
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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? – Maths Wizzard Oct 07 '22 at 21:05
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Please add the relevant information so that it does not rely on the image alone. – Arturo Magidin Oct 07 '22 at 21:06