Examples of Not Uniformly Continuous Function

Asked Dec 15 '20 at 01:45

Active Dec 15 '20 at 01:51

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I'm trying to practice proving that functions are not uniformly continuous using the $\varepsilon-\delta$ definition, but I'm unable to find much practice with it. Ones that I have tried so far are $x^3$ on $[1,\infty)$, $\frac{1}{x^2-1}$ on $(1,\infty)$, $x^2$ on $[0,\infty)$ and $\frac{1}{x}$ on $(0,1]$.

But I'm still uncomfortable and need more practice with this particular topic. Does anyone have more examples of simple functions that are not uniformly continuous over a given interval?

edited Dec 15 '20 at 01:51

asked Dec 15 '20 at 01:45

D Clark

Your last question is confusing. Which do you want? – herb steinberg Dec 15 '20 at 01:48
Take your functions and multiply them by 42. – Michael Dec 15 '20 at 01:50
I recommend you to search for some books that are made for the purpose of providing only counterexamples in real analysis. – B E I R U T Dec 15 '20 at 01:51
Sorry for the confusion. The question is edited – D Clark Dec 15 '20 at 01:51
2

Any unbounded continuous function on $(0,1)$ cannot be uniformly continuous. – zkutch Dec 15 '20 at 01:55
@Michael I suppose that you're correct that those are not uniformly continuous, but the proofs would look exactly the same. I'm looking for something new to try. – D Clark Dec 15 '20 at 01:55
How about $\frac{1}{x}$ on $(0,\infty)$? – B E I R U T Dec 15 '20 at 01:58
https://math.stackexchange.com/questions/2229195/ex-sin-x-is-not-uniformly-continuous?rq=1 – B E I R U T Dec 15 '20 at 02:07

Examples of Not Uniformly Continuous Function

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