Uniformly continuous functions in terms of limits

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In terms of limits, a function is continuous at a point $a$, if $\lim_{x\to a} f(x) = f(a)$. Now, what can we say about uniformly continuous functions in terms of limits?

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$f(x_n)-f(y_n) \to 0$ whenever $x_n-y_n \to 0$.