How can we define "curve" in elementary geometry (without using the notion of function)?

Question

Most of definitions of a curve are based on the notion of continuous function (https://en.wikipedia.org/wiki/Curve).

But continuous function is not an instrument to produce a continuous curve in general. The simple example $f(q) = q$ for all $q \in \mathbb Q$ shows us that a continuous function applied to a non-continuous domain may have a non-continuous graph.

Thus, we take an object that does not generate a continuous curve in general and apply it to another object, the real numbers. And how do we know the result is a continuous curve? Because we defined it this way.

This looks like a broken logic. I would expect the following logical steps:

1. We must have a definition of a continuous curve at least for some class of curves, at least in Euclidean 2D and 3D spaces;
2. Then we provide a definition of a continuous transformation and check if the result of such transformation is in agreement with the first definition;
3. Once we satisfied with the comparison, we can drop the first definition and take the second one as a generalization.

However, encyclopedia says: “In elementary geometry the concept of a curve is not clearly defined” (https://www.encyclopediaofmath.org/index.php/Line_(curve)).

I cannot believe mathematicians stopped trying to give a definition of a curve for elementary geometry (may I call it a Euclidean curve?):

• The definition should not use the notion of function or map;
• It should cover all primitive curves like intervals, arcs, etc.;
• It should not be a composition of primitive or predefined curves;
• It should be easily compared with the definition of a continuous function for verification.

Maybe, somebody knows examples of such definitions?
I am looking for a definition that can be shown in a high school, at least as an example.

Answer 1

My attempt (for a curve that may cross itself):
a Euclidean curve is a set of points in a Euclidean space $\mathbb R^n$ with the following properties:

1. It has a linear order $<$;
2. It is dense:
   • for any two points $A < A'$ of the curve, and any length $l > 0$ there are points $B < B'$ of the curve such that $A < B < B' < A'$ and $|BB'| < l$;
3. It is complete:
   • for every sequence of segments $|AA'| > |BB' > |CC'| > ... $, $A < B < C < ... < C' < B' < A'$, where the length of the segments converges to $0$, there is a limit point belonging to the curve;
4. We can measure length between any two points:
   • for any two points of the curve $A < A'$ there is an upper limit of sums of the lengths of consecutive segments $|AB| + |BC| + ... + |C'B'| + |B'A'|$, where $A < B < C < ... < C' < B' < A'$.

Definition: a limit point $X$ for a sequence of segments $|AA'| > |BB' > |CC'| > ... $, $A < B < C < ... < C' < B' < A'$ is a crossing point if $X < A$ or $X > A'$.

5. All crossing points of a curve are isolated:
   • for any crossing point $X$ there is a distance $d > 0$ such that for any crossing point $Y \ne X$: $|XY| > d$.