In a proof it was said that for a real value function $f$ that from $\mid f(x)-f(y)\mid \leq \mid x-y \mid$ it follows that f is continuous. Why is that?
Well in the problem the domain of the function was $[0,1]$ but it suppose it doesn't matter.
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5Do you know the definition of continuity? – Feb 20 '20 at 16:09
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Yes. Maybe is obvious but I don't see it :) – LIR Feb 20 '20 at 16:11
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Please see this. https://math.stackexchange.com/questions/2374289/understanding-lipschitz-continuity – You_Don't_Know_Who Feb 20 '20 at 16:11
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2$\delta =\epsilon$? – Peter Szilas Feb 20 '20 at 16:11
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Choose $\delta=\epsilon$ and the condition for continuity is met immediately. – CyclotomicField Feb 20 '20 at 16:12
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oh... that was easy... thanks – LIR Feb 20 '20 at 16:12
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@LazarIonutRadu Great. I don't want to use the word "obvious". So in simple terms, the definition says that "if $|x-y|$ is small, then we should be able to make $|f(x)-f(y)|$ small as well". Can you see how the condition $|f(x)-f(y)|\leq |x-y|$ guarantees that? – Feb 20 '20 at 16:13