What is the fundamental difference between a surface that diverges to infinity for a bounded, uncountable, 1-dimensional subset of its domain or rather when approaching the outermost limits of its open/bounded domain (and remains diverging/undefined outside the region bounded by that limit curve), and a surface that diverges to infinity when approacing the infinite limits of its open/unbounded domain (as in a paraboloid)?
Is the former surface somehow infinitely steeper than the latter?
>>8099604
Obviously the first one gets much steeper much faster. This about like asking the difference between [math]y=x^2[/math] and [math]y=\frac{1}{1-x^2}[/math].
>>8099604
>What is the fundamental difference between a surface that diverges to infinity for a bounded, uncountable, 1-dimensional subset of its domain or rather when approaching the outermost limits of its open/bounded domain (and remains diverging/undefined outside the region bounded by that limit curve), and a surface that diverges to infinity when approacing the infinite limits of its open/unbounded domain (as in a paraboloid)?
How correct is this sentence in terms of formal mathematical terms and definitions? Would you choose some terms differently when expressing what's being asked?
Do you like the thin manifolds? Aren't they somehow pleasant and interesting?
What the fuck is that thing you drew in the bottom right corner OP
Formally, how is the difference between these types of surfaces expressed?
What can I call the thin surface?
>>8099604
>diverges to infinity
This makes no sense. A surface is a collection of points, there is no notion of a "diverging surface". So hard to understand what you mean because you're not using nomenclature properly. You seem to believe that a function intrinsically defines its own domain, when it doesn't. A function is a triple consisting of its domain, codomain and a relation encoding where it sends its domain to.
I'm guessing you're asking for the difference between functions from subsets of the plane to the real numbers and how they diverge.
Firstly, you need to decide what class of functions you are looking at. For example meromorphic functions have isolated singularities, that is the set of poles is a discrete subset, (subsets where each point has a disjoint open neighbourhood.)
>>8100911
>Firstly, you need to decide what class of functions you are looking at
Wouldn't it have to be functions of the form z=f(x,y)?
>>8100970
z=f(x,y). That's an equation.
A general function of two real variables has no restrictions on it at all basically. For example I can define a function which has a singular set X just by setting it to infinity on X and 0 elsewhere. There is nothing in the definition of function which stops me from adding infinity to the codomain and then just sending all X to infinity.
I suggest you look up restricted classes of functions such as: continuous, smooth, analytic functions etc. From there you can research their limiting behavior.
They are not isometric. In particular, the open disk is not complete.
If you just take them as differentiable manifolds they are diffeomorphic. However, it's normal to require that embeddings of manifolds are "proper" which means that the inverse of compact sets are compact. Intuitively this just means if you go off to infinity in your manifold you should also go off to infinity in whatever you are embedded into. The open disk does not satisfy this, so it is not a "nice" embedding.
>>8101241
What would be some other examples of embeddings that aren't nice?
So when inspecting surfaces like this from a general viewpoint, it's useful to abandon representations as functions and just consider their properties as differentiable manifolds?
>>8101759
The most common example is embedding a line (which has open ends) souch that one of the ends comes back to almost touch another point of the line, getting infinitely close
>>8102243
As a Riemannian manfiold at least. Just going to differentiable throws away the metric which may be something you care about.
And on the other hand for surfaces embedded in 3 space there's stuff like the gauss map and morse functions where it is useful to think of having an embedding, but for general things you want to go more abstract
