# Why can't an electric field line suddenly break?

Electric field lines have no physical existence: they are useful concepts to understand vector fields, but they do not carry an independent physical ontology of their own.

Given a vector field $$\mathbf E(\mathbf r)$$, we define the field lines as the solutions of the differential equation $$\frac{\mathrm d\boldsymbol \gamma}{\mathrm ds} = \mathbf E(\boldsymbol \gamma(s)), \tag 1$$ i.e., as the continuous curves whose derivative is given by the electric field at the curve. (This is up to re-parametrizations of the curve, which turn the equation above into a proportionality, but which don't affect the geometric locus of the field line, which is ultimately the only thing we care about.)

The definition $$(1)$$ means that field lines can have kinks where their derivative is discontinuous if they meet places where the electric field is discontinuous, such as at a surface charge.

However, field lines cannot break because we define them as continuous objects: basically, putting a pen at a starting point and then following the arrows of the vector field without lifting the pen.