an electrostatic field lines cannot be discontinuous. why?
Electric field lines can be discontinuous! Just, they are never discontinuous in charge free space. At positions of charges, you draw discontinuous field lines.
Gauss's law in its integral form says:
& int;D·dA = Q
Here Q is the charge enclosed in an arbitrary closed surface, ∮ is the integral over the closed surface and D is the electric displacement field. dA is the surface measure and the ∫ is a contour integral.
∫D·dA is the number of all field lines leaving the region minus the number of all field lines entering the region.
If there is no net charge Q = 0. Therefore ∫D·dA = 0. This means, in simple words: the number of field lines leaving the region through the closed surface is the same as the number of field lines entering the region.
So, if you draw a field line that ends somewhere, you will define a surface that encloses the end of the field line. Then the number of field lines entering the region is not the same as the number of field lines leaving the region. Thus, this is only possible if Q ≠ 0.