Is it always true that for any given curve and for any two points on that given curve (no matter how close these points are), it is possible to construct some curve that intersect that given curve only at those two points (with the below constrictions) ?
Constrictions:
1- The other curve ( the constructed curve ) should not be a straight line in a neighborhood of the two points .
2- The two curves should be continuous in a neighborhood of the two points.
3- The two curves should be differentiable in a neighborhood of the two points.
As an example take the parabola $ y = x^2 $ and for a given $ε>0$, let ($x$ , $x^2$) and ($x+ε$ , $(x+ε)^2$) be two points on the parabola $ y = x^2 $. My question is : is it possible to find a curve that passes through only those two points for any given $ε>0$ ? And how to construct such curve ?
If this is always possible please provide an explanation. And if not (please provide a counter example) how about removing constraint 3 dose that make it possible ?
Edit: If the question is too broad, then It would be helpful to provide a specific example for the case of a parabola.