$(a)$ An electrostatic field line is a continuous curve. That is,a field line cannot have sudden breaks. Why not?
$(b)$ Explain why two field lines never cross each other at any point?

(N/A) An electrostatic field line is a continuous curve because a charge experiences a continuous force when traced in an electrostatic field. The field line cannot have sudden breaks because the electric field intensity varies continuously in space,and a test charge would move along a continuous path without jumping from one point to another.
$(b)$ If two field lines were to cross each other at a point,it would imply that the electric field intensity at that specific point has two different directions simultaneously. Since the electric field is a vector quantity and must have a unique direction at any given point,this is physically impossible. Therefore,two field lines never cross each other.