I want to show that the piece wise function $f(x)=x$ for $0\le x<1$ Or $f(x)=x^3$ for $1\le x\le 2$is uniformly continuous. I can show uniformly continuous on each interval separately but how do I show uniform continuity on of the whole function on the entire interval.
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Every continuous function on a compact interval is uniformly continuous. – amsmath Oct 22 '21 at 02:11
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What if I’m not familiar with this theorem? Can I show from definition? – Danny Oct 22 '21 at 02:32
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You do not have to use the definition. Observe that $f$ is continuous map on a compact (or closed and bounded) interval to the real numbers, hence, it must be uniformly continuous. If you are not familiar with this result, you may want to take to a look here.
ashK
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How do I show using definition or at least intuitively for a beginner like me? – Danny Oct 22 '21 at 02:33
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@Danny If you are trying to prove this by using the definition, you must appeal to Completeness Axiom of some form of other. – ashK Oct 22 '21 at 05:35
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@amsmath I am referring to the $\epsilon-\delta$ definition. Of course, I can easily use this definition to show uniform continuity piecewise but how do I do this for the function on the entire interval? – Danny Oct 22 '21 at 16:27
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1We told you how. Look up the corresponding theorem and its proof. Of course, the $\epsilon-\delta$ definition is used in the proof. – amsmath Oct 22 '21 at 16:35