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In trying to explain irrational numbers, I am now wondering how to explain continuous curves. (Related to limits and such).

In discrete systems (topology, geometry, etc.), a curve makes sense because you go from point to point $a \to b \to c \to \dots$. But in trying to explain something like a continuous circle, you can say "first start off with a polygon say 10 sides, then 11, ... then 100, then 1000, etc. Eventually the lines between points are so small to be infinitely small (or infinitesimal). That's when you get a perfect circle, etc."

You can also talk about parabolas, and there being an "instant" a parabola changes from direction $a$ to direction $b$. But if it is continuous, then I don't see how there will ever be a point at which it switches directions. So it seems there must be discreteness at some point.

With the circle, if every edge was infinitely small then it would take infinitely long to just traverse a single edge or something like that. So I start getting confused.

Wondering how to explain how these properties work to a layman. How can an infinitesimal change allow for any movement (since it is infinitely small), so we can traverse the curve. And wondering if there is anything the discrete areas of mathematics says about this problem.

Lance
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    You're sort of getting into Zeno's Paradoxes here, https://en.wikipedia.org/wiki/Zeno%27s_paradoxes, maybe you'd be interested in reading more – Badam Baplan Jun 20 '18 at 14:33
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    Back in the old days, we were taught: "a continuous curve is a curve which you can draw on a piece of paper with a pen without the pen leaving the paper". But wait, do students still know what a pen is these days? – achille hui Jun 20 '18 at 14:39
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    Also a parabola will never really switch from direction $a$ to direction $b$ in an instant, since the derivative of a parabola is itself continuous. – Badam Baplan Jun 20 '18 at 14:50
  • How do you, personally, get from point $a$ to point $b$ on any given day? – Badam Baplan Jun 20 '18 at 14:53
  • I would challenge the idea that a parabola even has a direction. Maybe you mean slope, in which case the layman's definition of slope should suffice. This also helps address your approach to defining the circle as a limiting process of polygons, since the slope of a curve is also a limiting process of line segments. Otherwise if you want to talk about something changing direction, you're really talking about a particle following a continuous path; in this case you're really talking about differential equations. – Sort of Damocles Jun 20 '18 at 15:10
  • Everyone has picked up a pen or pencil and drawn all sorts of curves with it. I've never met anyone who had trouble with that idea. What you've got to explain to them the concept of discontinuity. Your problem is not with explaining a continuous curve, but rather with explaining an unnecessary concept that they are somehow polygons with infinitesimal sides. (I was going to say "false concept", but have to admit you can view them in this fashion. It just isn't helpful.) – Paul Sinclair Jun 20 '18 at 23:03

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Infinitesimals provided Cauchy with the original way of understanding continuity, according to the idea that infinitesimal change in $x$ must always lead to an infinitesimal change in $y$. In more detail, a function $y=f(x)$ is continuous, according to Cauchy's definition in Cours d'Analyse, of an infinitesimal change $\alpha$ of the independent variable $x$ always leads to an infinitesimal change $f(x+\alpha)-f(x)$ of the dependent variable $y$. In teaching practice this definition is more effective than the epsilon-delta one, as argued in this 2017 publication in Journal of Humanistic Mathematics.

As to your question "How can an infinitesimal change allow for any movement (since it is infinitely small), so we can traverse the curve" your concern seems to be that making finitely many infinitesimal steps can only lead you an infinitesimal distance away from the starting point. This is correct, but the point is that after an infinite number of steps one does get an appreciable distance away from the starting point. This was the original viewpoint of Leibniz who thought of a circle as an infinite-sided polygon. Similarly, an integral would be defined via an infinite (hyperfinite) partition of the interval of integration; see Keisler's textbook for details.

Mikhail Katz
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    Why is this answer downvoted? I found it to be clear and useful. +1 –  Jun 22 '18 at 07:47
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    @Brahadeesh, I have often wondered about that, as well as numerous other downvotes on questions under the tags [tag:infinitesimal] and [tag:nonstandard-analysis], etc. It seems some editors are just not comfortable with the idea that infinitesimals could be more useful than epsilon-delta. It seems to touch their belief system. – Mikhail Katz Jun 22 '18 at 08:00
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    that’s...unexpected! Please do continue contributing to this site, your answers are beneficial to a lot of us, myself included. :) –  Jun 22 '18 at 08:37