It seems to me (who is quite the math novice) that a very important ‘statement’, for a lack of a better word, that is foundational to many mathematical topics is that a given curve, which is continuous and differentiable, can be built from a bunch of straight lines as long as we make those lines ‘small enough’. For a 2D case, I interpret this as being able to build a curve that traverses through a 2D plane by only using little $\Delta x$ ’s and little $\Delta y$ ’s. I am wondering how one goes about proving this statement. It seems to me a good starting point can be illustrated using the following picture:
I suppose I should clarify that I am simply using this circle as a starting point for this argument...this could be any arbitrary curve (not just the circumference of a circle...though I suppose there is probably a proof out there that shows tiny sections of a curve can also be approximated by an arc length of a circle with a certain radius...but that's another question for a different time).
So the question I want an answer to is the following: As $\Delta x$ becomes very small (and its corresponding $\Delta y$, based on the behavior of the curve, or, more specifically, based on the function that describes the curve, also becomes very small ), does
$(r*\Delta \theta) / (\sqrt{(\Delta y)^2+(\Delta x)^2)}$ approach 1.0?
How would one go about proving this? I feel like most arguments that I can think of are rather circular…in that I have to use a property that is based off of what I want to prove in order to prove it! Is there a proof for this limit? Or is this just an axiom we accept to be true?
Edit 1: It has been brought to my attention that including the word "differentiable" as a characteristic of a curve creates a circular argument for what I would like to prove. The logic behind that claim is "if the curve is differentiable, then of course a curve can be decomposed into line segments because that is the definition of differentiable". Assuming this is true, please disregard the word 'differentiable'. I am interested in solving the previously referred to limit as if I never knew that calculus existed!
This is true for most "nice" curves but isn't true in general, at least under the usual definition of a linear continuum, particularly for space-filling curves as @PeterShor mentioned in their comment. If a space-filling curve could be approximated by straight lines, as I understand your condition, it would be differentiable everywhere, and there exists no space filling curve that is differentiable everywhere (though it is possible for it to be differentiable almost everywhere, see https://en.wikipedia.org/wiki/Sard%27s_theorem and https://mathoverflow.net/questions/201424/proof-that-no-differentiable-space-filling-curve-exists for a discussion of this.)
The notion of continuity is actually a bit tricky and relies on some general topology. For all intents and purposes in introductory calculus, this is true. I think this should be true for any curve that can be represented with a Taylor series. But it is not true for any arbitrary curve.