How to explain a continuous curve to a layman 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.
 A: 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.
