How do curves consist of points? According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in other words there is nothing, then how is it possible to draw any curve? 
The thing I could imagine is that maybe a point is not as Euclid thought. I mean a point can be thought of as a small line segment whose length is approaching zero but never becomes exactly zero. In this way, we can say that curves consist of points (the line segment with length approaching zero). But then it breaks the fact as given by Euclid that a point has no dimensions.  Please help me to get out of this dilemma.

A point in an n-dimensional Euclidean space is that point which consists of n coordinates which give a distinct location of that point. 
A function in Euclidean space essentially provides a rule for defining a set coordinates in space for which that function is true. 
You are correct in that the physical realization of a collection of 0-dimensional points is nonsense to create a physical 1-D curve, however, it makes tons of sense when you learn a bit more math in topics like set and measure theory
