Help with choosing delta-epsilons regarding continuity

Question

Is there any good strategy for picking an epsilon-delta in local and global continuity? I'm really struggling to pick such values. For example, I'm not sure entirely how the author got the values stated in the excerpt:

Answer 1

In general, for continuity you can just compute $|g(x + \delta) - g(x)|$ and see if you can find a "small" upper bound in terms of $\delta$, i.e. such that the bound goes to $0$ as $\delta \to 0$. If you can, then you've shown continuity. For $g(x) = 1/x$ you get $g(x)- g(x + \delta) = 1/x - 1/(x + \delta) = \delta/x(x + \delta)$ which you can show goes to $0$ as $\delta \to 0$ for fixed $x \neq 0$ so you have continuity of $1/x$ for $x \neq 0$. However you do not have uniform continuity, because for that you would need a uniform bound of $|g(x)- g(x + \delta)|$ for fixed $\delta$ that works for every $x$, and this is not possible for $g(x) = 1/x$.