Question

why can't electrostatic field line be discontinuous?

Answer 1

These lines are assumed to be continuous and travel to infinity because their discontinuity would mean that the charge has suddenly vanished & reappears again which is illogical 

Answer 2

Answer:

• An electrostatic line is a continuous curve, because tangent to it at any point represents the direction in which a test charge kept at that point will experience force.
• It cannot have sudden breaks, because no abrupt force acts on a test charge.