Lines cannot be discontinuous because if it were discontinuous at a point in space it would mean there is a sink of the charge there, but we know that charged cannot be created or destroyed so there must be some value of the field set up by a charge at every point in the space.

Electric field lines cannot be discontinous. This is becoz Elec. field is inversely proportional to square of distance . Hence it can be zero only at infinity & cannot end abruptly. Since tangent to elec. field lines represent elec field at that point , so elec field lines can't have breaks.

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Arti Princess answered this

In simple words v can say dat if any charge is producing electric field around it then it cant b discontinuous.. because this cant happen that electric field is there at a point then it breaks then it is again found at another point.. electric field shud b continous..