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We say $f$ is uniformly continuous on a set $D$ if

\begin{align*} \forall \epsilon >0, \exists \delta >0 , s.t.\\ \forall x, a \in D, |x-a| \le \delta \rightarrow |f(x)-f(a)| \le \epsilon \end{align*}

Is this definition equivalent to the following:

\begin{align*} \forall \epsilon >0, \exists \delta >0 , s.t.\\ |x-a| \le \delta \rightarrow \sup_{x, a \in D}|f(x)-f(a)| \le \epsilon \end{align*}

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The second definition uses the notation 'sup' meaning supremum or least upper bound. It is quite possible to define uniform continuity using this notation, for example:

\begin{align*} \forall \epsilon >0, \exists \delta >0 , s.t.\sup_{x, a \in D s.t. |x-a| \le \delta }|f(x)-f(a)| \le \epsilon \end{align*}

This is equivalent to the first definition. However, in the second definition the expression $|x-a| \le \delta$ is neither within quantifiers over $x$ or $a$ nor within the scope of 'sup'. Taken strictly, there is nothing in the second definition to say that the $x$ and $a$ in that expression belong to $D$. So although the second definition is not incorrect or inconsistent with the first definition, its use of notation is not as clear as it could be.

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They are the same, because if the 'sup' (an upper bound, specifically the least upper bound) of $|f(x) - f(a)| < \epsilon$ as in your definition, then all $|f(x) - f(a)|$ must be less than $\epsilon$, which is equivalent to the definition of uniformly continuous.

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