(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 ?

Explanation

(a) Electric field, surrounding to any charge system is found to be existing at every point surrounding to it with definite magnitude and definite direction up to infinite region in three dimensions. And we know that presence of electric field is depicted pictorially by electric field lines. Hence, if electric field exists everywhere surrounding to charge system then electric field lines also must be defined which can end up only at infinity. Till then, they have to continuous. They cannot have sudden breaks in between.
(b) Electric field line is that characteristic curve the tangent drawn at every point on which gives us direction of electric field at that point. Hence, two or more electric field lines cannot intersect at any point in an electric field because if they do so then electric field would exist at that point in more than one directions, along more than one tangent drawn at that point, which is not possible.