I need help proving that $\frac{1}{x+2}$ is not uniformly continuous for $(-2,0]$, using an $ε-δ$ proof.

I understand that intuitively this is just the function $\frac{1}{x}$ which is not uniformly continuous on $(0,1)$ shifted 2 units to the left, but I am struggling as to how to write a technical proof.

Specifically, I am unsure what $f(x)$ and $δ$ I should use to prove that $|f(x)-f(y)|>ε$ when $|x-y|<δ$.

  • $\begingroup$ Isn't a uniformly continuous function on a bounded interval necessarily bounded? $\endgroup$ – Angina Seng Apr 26 at 19:34
  • $\begingroup$ What is the definition of uniform continuity and how is it different from standard continuity? $\endgroup$ – Doug M Apr 26 at 19:40
  • $\begingroup$ @DougM : The diffference is that for any given $\varepsilon,$ there is a suitable $\delta$ that is the same everywhere within the domain instead of being different in different parts of the domain. $\endgroup$ – Michael Hardy Apr 26 at 19:43
  • $\begingroup$ You need to prove that $ \exists \epsilon : \forall \delta, |x-y| < \delta \wedge |f(x)-f(y)|>=\epsilon$ $\endgroup$ – user6767509 Apr 26 at 19:55

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