Question

why electric field line is continuous curve in nature

Answer 1

The field lines appear to be parallel straight lines. Of course, they are continuous and as they appear to be parallel, their magnitude is nearly the same. The idea is that electric field lines can start on positive charges and finish on negative charges. In such a case electric field lines would be discontinuous.

Answer 2

Continuous curve means, a field line cannot have sudden breaks. Electric field lines exist throughout the region of an electric field. The electric field of a charge decreases gradually with increasing distance from it and becomes zero at infinity i.e., electric field cannot vanish abruptly. So electric field line cannot have sudden breaks, it must be a continuous curve.