Uniform Continuity on Compact Sets and Cauchy Sequences

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We prove that every continuous function on a compact set is uniformly continuous.

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Sir, If a function is uniformly continuous on a closed interval, could we refine the definition of uniform continuity by replacing the condition |x-y| < δ and |f(x) - f(y)| < ε with |x-y| ≤ δ implying |f(x) - f(y)| ≤ ε ?

sumittete
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But when convergence point is not in compact set non uniform function will also be continuous in compact
I can say y = x^2 is continuous in compact set [1, 5] so it should be uniform function....won't this be a contradiction to the theorm?

angadbhatti