If we have a uniformly continuous function, and we know that there exist $\displaystyle \delta$ s.t.$\displaystyle \mid f(x)-f(y)\mid<\epsilon$ $\displaystyle \forall x,y$ s.t. $\displaystyle \mid x-y\mid<\delta$, does this mean that $\displaystyle \forall x,y$ s.t. $\displaystyle \mid x-y\mid\le\delta$ that $\displaystyle \mid f(x)-f(y)\mid\le\epsilon$?

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