A b-metric is defined similar to a metric in which the triangle inequality is replaced by the inequality $$d(x,z)\leq s\Big[d(x,y)+d(x,z)\Big]\quad\forall\ x,y,z$$ where $s\geq1$.

There is an example of a b-metric which is not continuous.

My question: If the b-metric space is complete, can we conclude that the b-metric is continuous?

Any help would be appreciated. Thanks!