# Show that $\frac{1}{x+2}$ is not uniformly continuous for $(-2,0]$.

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 $$|x-y|<δ$$.

• Isn't a uniformly continuous function on a bounded interval necessarily bounded? – Angina Seng Apr 26 at 19:34
• What is the definition of uniform continuity and how is it different from standard continuity? – Doug M Apr 26 at 19:40
• @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. – Michael Hardy Apr 26 at 19:43
• You need to prove that $\exists \epsilon : \forall \delta, |x-y| < \delta \wedge |f(x)-f(y)|>=\epsilon$ – user6767509 Apr 26 at 19:55