Is there any difference between them? or are they same? I was going through the definition of uniform continuity, which uses the term "for every". Can someone explain me the difference between the two terminologies. Please try to explain it in the context of uniform continuity definition. Thanks!
2026-03-26 21:27:45.1774560465
What is the difference between "for every $x$ there exists a $y$" and "for all $x$ there exists a $y$"?
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The way you have stated it in your title, the two statements are equivalent.
But you probably wanted something different.
For every $x$ there exists a $y$ means that $y$ depends on $x$ and may as well change with $x$.
For example for every $x$ there exists a $y$ such that $y=x^2$ (that is a true statement )
On the other hand there exists a $y$ such that for all $x$, $y=x^2$ is false.
Thus you may not switch the different quantifiers around.