electric-fields
In a charge-free region, electric field lines can be taken to be continuous curves without any breaks
What is meant by charge-free region?
I suppose it means absence of any external charge.
What if some other charge is present near the source of the field, will it make the field lines discontinuous?
Why does it state that "field lines can be taken to be continuous"? Are they not exactly continuous?
Then will every field line originate from the +q and end up to -6q
No, every field line won't end to negative charge.
will there be some extra lines coming to -6q from infinity because of higher charge to get 6 times the number of field lines?
Yes, many extra lines will come.
I think it should be that every line will originate from the positive and go to the negative, only difference will be in the density of the field lines. Am I right?
No, you are not right. Only some of field lines will end in -6q from +q.Yes, there will be a change in density gradient around charges.
why can I say that the flux near +q will be equal to that near -6q?
Flux around two charges would be different. You can simply use Gauss law for flux.
I would take it to mean that, for the Electric field $\overrightarrow{E}$.
$$\overrightarrow{\nabla}\cdot\overrightarrow{E}=0$$
Everywhere within your charge-free region (lets call it M). Intuitively you can think of a point charge as a place where all of the electric field lines begin or end. picture the field as several arrows, they will point toward or away from the charge, this is what the divergence in the above equation measures.
You could also consider (applying Gauss's Law to the above) to be:
$$\int_{\partial M}\overrightarrow{E}\cdot d\overrightarrow{a}=0$$ where $\partial M$ denotes the boundary of the charge free region. Though this is less helpful as it really means the total charge is zero in M (so you could have charges here so long as they sum to zero).
Best Answer
By convention, electric field lines are said to start from a positive charge and end at a negative charge. As you can see in the above figure, the field lines come to an abrupt stop at the surface of the charge. When there isn't any charge, the electric field lines must be continuous. The only place where they can start or end is at a charge.
Bonus: Fields tend to be uniform at points far away from the charge source
In the above figure, consider that the big black circular patch is a charge. As you move away from the charge, the electric field lines not only tend to be continuous, they also have nearly the same strength.
For example, have a look at the region bounded by the red rectangle. 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.