An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?

An electric field line always points in the direction that a positive test charge would accelerate in if placed upon that line. A charge will experience a continuous force when placed in an electrostatic field. Therefore, an electrostatic field line will be a continuous curve. The field lines do not have any sudden breaks as the test charge or any charge will move continuously and does not jump from any point to another.