I was reading the Wikipedia article on (uniform) equicontinuity, and it states
This [equicontinuity] criterion for uniform convergence is often useful in real and complex analysis. ... In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the mean value theorem or some other kinds of estimates can be used to show the sequence is equicontinuous. .... A similar argument can be made when the functions are holomorphic. One can use, for instance, Cauchy's estimate to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic.
So I'm aware that if a sequence of real differentiable functions $\{ f_n(x) \}$ on a compact set $[a,b]$ has a uniformly bounded derivative, then the family is equicontinuous (actually it's equi-Lipschitz which is stronger). I also realize that pointwise equicontinuity implies uniform equicontinuity (actually, they're equivalent) for continuous functions on a compact set. However, I don't know how to apply the ideas mentioned in the Wikipedia article. Can anyone expand and give examples on how we use the conditions mentioned above, specifically:
- How do we use regularity (which I'm taking to mean the degree of differentiability of the functions) to prove equicontinuity?
- Can we use Cauchy estimates (which I've only seen in the context of complex functions) to prove equicontinuity of real functions? If so how?
- What are some other conditions/theorems that imply equicontinuity which don't require uniform convergence as a condition (since that's what you're using equicontinuity to prove)?