Proving electric lines of forces can't be discontionous in charge free

[namanjain]

Hey guys
i had a question in my paper
why electrostatic field lines cannot be discontinuous in charge free region

i guessed a weird (but an innovative proof)
Tell me is it correct

So here it goes
"Let's assume that ELOF can be discontinuous

Then i draw a diagram of broken electric field
Now at one of the two free ends i assumed a small Gaussian volume(Only the free end)
Now using gauss law
ø:FLUX
ø = ∑Q_enclosed/ε
ELOF ARE ENTERING BUT NOT ESCAPING SO ø≠0
BUT ∑q_ENCLOSED=0
SO OUR ASSUMPTION IS FALSE
H.P."

 

[Simon Bridge]

Gauss' law sort-of assumes that field lines cannot be discontinuous: so, by invoking it like that you are assuming the statement you are supposed to be proving.
Note: you don't need more than one free end in the hypothetical diagram.

The property is more part of the definition of "field line" so it is not something you prove exactly.
I suppose you could look at the consequences in terms of a small test charge... it travels along a field line to the end, then what?

You describe the Gauss' law argument as "novel" - have you been provided with a different "proof"?