For an additive κ \kappa -metric space X X with an s ( x ) s\left ( x \right ) -continuous κ \kappa -metric d ( x , C ) d\left ( {x,C} \right ) , we prove that X X is metrizable, and that if d ( x , C ) d\left ( {x,C} \right ) is locally regular, then z ( x , y ) z\left ( {x,y} \right ) is bicontinuous, and ρ ( x , y ) = z ( x , y ) + z ( x , y ) \rho \left ( {x,y} \right ) = z\left ( {x,y} \right ) + z\left ( {x,y} \right ) is a metric on X X which agrees with the topology of X X .