Is there some trick/formula to finding your delta value when proving continuity and/or uniform continuity? This seems to be very hit or miss for me, and I find myself often finding having a lot of trouble with it.
For instance, if we have $f(x) = \frac{1}{x^2 + 1}$, and want to prove it's uniformly continuous from $\mathbb{R} \rightarrow \mathbb{R}$, apparently the delta value that's useful is $\epsilon / 2$. But, I just cannot see how to get this value.
Any help would be greatly appreciated. Thank you.
For a differentiable function like that, one trick is to see if you can bound the derivative as less than some constant $C$. Then you can find a good $\delta$ since
$$|f(x) - f(y)| \leq C |x - y|$$
So, if you want $ |f(x) - f(y) | < \epsilon$, then you can just choose $\delta < \frac{\epsilon}{C}$.