According to the definition:
$f$ is $R$ integrable on $[a,c]$ if and only if for every $\epsilon>0$ there exists a partition $P$ such that $U(P,f)-L(P,f)< \epsilon $
Here, does the definition say that:
"there is a particular $P$ which makes the statement hold for every $\epsilon$"?
$$OR$$
does the definition mean:
"for any particular $\epsilon$ there is a particular $P$ for which the statement holds"?