the continuity equation
Charge is never created or destroyed, and — more strongly — it is never even teleported: if the charge in a region changes, exactly that much charge must have flowed across the region's boundary. The continuity equation is the mathematical statement of this local charge conservation.
In differential form it reads d rho/dt + div J = 0, where rho is the charge density and J the current density. Read it as a balance sheet for a tiny volume: the rate at which charge density falls (minus d rho/dt) equals the net current flowing out (div J). Integrated over a finite volume it becomes dQ/dt = minus the flux of J through the bounding surface — the enclosed charge decreases exactly at the rate charge streams out through the walls. The word local is the whole point: charge does not merely stay constant globally, it moves continuously from place to place, never vanishing here to reappear there.
In electromagnetism the continuity equation is not an independent postulate; it is forced by Maxwell's equations. Take the divergence of the Ampere-Maxwell law and use Gauss's law and you recover d rho/dt + div J = 0 automatically — indeed this is exactly the consistency that Maxwell's displacement current was invented to guarantee. In relativistic notation it collapses to the single line d_mu J^mu = 0, the vanishing four-divergence of the four-current.
A small sphere holds charge Q that is leaking away through a wire at 3 amperes. The continuity equation says dQ/dt = -3 C/s: the charge inside falls at exactly the rate current carries it out across the surface. There is no way for Q to drop without a matching current through the boundary.
d rho/dt + div J = 0: charge lost from a region equals charge flowing out across its boundary.
Continuity expresses LOCAL conservation, stronger than merely 'total charge is constant' — it forbids charge disappearing at one point and reappearing at another. In electromagnetism it is a theorem, not an axiom: it follows from taking the divergence of the Ampere-Maxwell law.