Re: Lipschitz continuity, I suggest you look at Fig. 29 in Arnold (1992) and the discussion around it. Myself, I think about the Lipschitz property as more of a weakened smoothness condition than a strengthened continuity one; the symbol $C^{1-}$ sometimes used for Lipschitz-continuous functions is quite apt. (Note that $C^0 \supset C^{1-} \supset C^1$; about the $C^k$ notation, see the MathWorld article in case you’ve never taken multivariable calculus.)
Re: other notions of continuity, I’m not sure that you are going to need them all. To enumerate:
- Continuity, I expect you have some understanding of.
- Uniform continuity is the same as continuity when you are on a compact set (the Heine-Cantor theorem). Uniform continuity is what you use to prove Riemann integrability, so a simpler version and applications of this result usually occur somewhere around there in a calculus/analysis course.
- Absolute continuity mostly appears in connection with Lebesgue integration, so that is where you should go for the explanations. Whether you need to depends on the notion of integration used in the ODE course, and if you indeed need to learn a whole new theory of integration then you’ve got bigger problems than continuity notions.
- Lipschitz continuity I tried to elaborate on above, and it is indeed the notion used for the usual ODE existence and uniqueness theorem, the Picard-Lindelöf theorem.
- Hölder continuity is a refinement of Lipschitz continuity (look at the definition) used mostly in advanced PDE theory. If you made it to fourth year without real analysis, you will be able to graduate without ever encountering it.
- Equicontinuity (and uniform equicontinuity, and Lipschitz property with a common constant) is a property of families of functions rather than of functions themselves. You should compare it to uniform convergence of functional sequences. It is used in the Arzelà-Ascoli theorem, which you may or may not end up using in the course.
Aside from continuity, your remarks make me wonder whether you took some kind of multivariable differential calculus. If not, study it, because it is required here. A ball there is just a normal Euclidean ball (possibly $n$-dimensional though), the excercise is mostly to figure out where in the usual calculus proofs you can change a modulus sign into a norm (i.e. length) sign, and where you should replace multiplication by a linear map (i.e. matrix multiplication). Functional (infinite-dimensional) balls should come afterwards.
Re: the course, brace yourself, because the proofs are going to be hard for you. Thankfully, there is not a lot of theorems in a basic ODE course, and the statements are all rather intuitive. Given that you are asking the question at this time of year, I suppose that the course began with the theorems, which is really not a wise choice. (However, if it is really all like this, i.e. if it is going to proceed into PDE theory, dynamical systems and so on, I strongly suggest you drop it.) Read the first chapters in Arnold’s book to get what you are actually getting at with those statements. (I would even recommend to track down a translation of the first Russian edition, because it is much more concise if sparser on applications.)
