If we choose any point of a line, a parabola, a circle, a sinusoidal curve etc, as a center of a circle of radius r, then there is a distance d such that, if r<d, the circle intersects the curve at exactly two points. I wonder if this is true for any plane curve that does not intersect it self. Fractal curves come to mind, that may provide a counter example, but I can not prove it. Space-filling curves also come to mind, but I am no sure they fulfill the definition of a simple curve (not self-intersecting). In any case, if we exclude space filling curves, are there any examples of simple plane curve, either closed, or open that tends to infinity from both directions, which has points such that, any circle around them, however small, could intersect the curve in more than two points?
Edit: What about fractal curves like Koch snowflake, Cesàro fractal, Fibonacci word fractal, Minkowski sausage etc.
This is true for a $C^1$ curve with non-zero derivative at all points. This comes from the fact that such a curve is locally diffeomorphic to a segment. This can be proven using the inverse function theorem or the constant rank theorem.
This is false for $C^0$ curves, or even for $C^1$ curves with a point with zero derivative. The curve may spiral around the point with zero derivative. Let make the spiral a bit square, and since a circle intersects a square in four points when they have the same centers and close diameter, you will have more than two intersection points. Here is a rough sketch. with the first bold curve spiraling to the point of zero derivative, and the lighter curve going away from it. Both meet at a point where the derivative is zero.
Finally, it can be shown using the Baire category theorem, that a generic continuous curve that intersect any segment not restricted to a point will intersect it infinitely many times. Such curve will also intersect a circle infinitely many times. This is a bit unintuitive if you have never heard of the Baire category theorem.