Can anyone provide a "hierarchy of functions/function sets" in terms of differentiability?

Question

I'm studying Advanced Analysis II, specifically differentiation in higher dimension vector spaces. I'd like to fully understand the logical implications and relations between differentiable, continuous and differentiable, only continuous functions, those who admit partial derivatives without being continuous, etc... You get the idea. For now, I only have the following hierarchy: $$C^0 \supset C^1 \supset ... C^\infty $$

I'd like to complete it and see where all the above mentioned functions (and more) lie.

Please notify me if the question isn't clear enough.

See my comments to Generalization of differentiability/smoothness class "$C^n$" notation. See also Notation of analytic functions $\mathcal{C}^{\omega}(\mathbb{R}^n)$. Incidentally, there are MANY different differentiability notions -- see my answer to Is there a garden of derivatives? -- and so some books and papers introduce various specialized notion to describe hierarchies based on other differentiability notions. — Dave L. Renfro, Apr 29 '22 at 17:14
@DaveL.Renfro Thanks, but as I said, I'm studying Analysis II. I'm asking about the logical implications between continuity and differentiation, not every kind of differentiability. — sleepydrmike, May 02 '22 at 12:51
I think you need to clarify your question, since "I'd like to complete it and ..." is not very informative. It sounds like you want a space that includes all functions in these spaces. In addition, having a presumably natural language use of the word "complete" muddies things a bit since the (metric) completeness of these spaces is one of their more significant attributes. Then in your comment you want logical implications between continuity and differentiation, which seems a bit below "Analysis II", as one learns in basic calculus that differentiability implies continuity but not conversely. — Dave L. Renfro, May 02 '22 at 14:41
If you're interested in the interplay between unrestricted continuity and differentiability and their various restricted versions (in a coordinate direction, along one or more specified lines, along one or more specified curves, etc.), then the situation is extremely complicated. See this 4 June 2005 sci.math post (coordinate wise and path-wise continuity relative to continuity, (continued) — Dave L. Renfro, May 02 '22 at 15:03
my comment to Set of possible limits of two-variable functions along curves -- connected?, my answer to Multivariable limits, my comments to Why are so many online sources "wrong" about directional derivatives? (pathologies with ordering of mixed partial derivatives). Also, Higher Order Derivatives by Mukhopadhyay (2012) is a virtual encyclopedia of highly technical issues. — Dave L. Renfro, May 02 '22 at 15:03

Can anyone provide a "hierarchy of functions/function sets" in terms of differentiability?

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