Is the function $f(x) = \frac{x+1}{x+2}$ uniformly continuous on $(-2,\infty)$? I know how to prove this for the case when the domain is closed or is, say, $[-1, \infty)$, but I am not sure how to estimate the fraction from above when working with this domain.
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1$\begingroup$ If it were uniformly continuous, what could you say about $f$ on the interval $(-2,0]$? $\endgroup$– Thomas AndrewsCommented Apr 12, 2017 at 20:44
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$\begingroup$ @ThomasAndrews, then $f$ is bounded on that interval? $\endgroup$– AldersonCommented Apr 12, 2017 at 20:48
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$\begingroup$ So when $x$ goes to -2, $f$ goes to $\infty$, hence, $f$ is not uniformly continuous. Is my reasoning correct? $\endgroup$– AldersonCommented Apr 12, 2017 at 20:50
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$\begingroup$ Yes, uniformly continuous functions must be bounded on bounded intervals, and $f$ is not. $\endgroup$– Thomas AndrewsCommented Apr 12, 2017 at 21:56
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1 Answer
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Consider taking two sequences: ${x_n}= -2 + \frac {1}{n}$ and ${y_n}= -2 + \frac {1}{2n}$, from there argue by contradiction.
