Does there exist a discontinuous metric?
Let $(X,d)$ be a metric space. If we consider $X\times X$ as a metric space with the discrete metric, then $d:X\times X\rightarrow\mathbb{R}^{\geq 0}$ is evidently continuous. It is possible to define a metric on $X\times X$ such that $d$ is not continuous with respect to that metric?
