Continuity (mathematics) in the context of "Charge density"

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

Like mass density, charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position ${\displaystyle {\boldsymbol {x}}}$, like a fluid, and ${\displaystyle \rho ({\boldsymbol {x}})}$, ${\displaystyle \sigma ({\boldsymbol {x}})}$, and ${\displaystyle \lambda ({\boldsymbol {x}})}$ are usually regarded as continuous charge distributions, even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge, the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density ${\displaystyle \rho ({\boldsymbol {x}})}$ and the current density ${\displaystyle {\boldsymbol {J}}({\boldsymbol {x}})}$.