On the tangent of a geometric curve

Question

If a function has it`s first partial derivatives continuous then if on this curve we fix two distinct points and let them join by a secant line then we know that when these point begin to approach each other then the portion of the curve between these two point is the same line as their secant when these two points are in a differential part of curve.But even in this differential part there are still infinite number of points.So should we conclude that tangent on such kind of curve (defined by a function of at least first order continuous derivative) touches more than one point of the curve?

Answer 1

No, because the tangent is defined as the limit of the secant through two point as the two points approach each other. The concept of limit is often not easy to grasp, and a study of its historical development will show you why.