Now, with respect to $d^*$ metric on the product metric space i will now try to prove the continuity:
Pick an arbitrary $\epsilon >0$ and consider the inequality $|d(x,y)-d(a,b)| < \epsilon$. Now, the metric function will be continuous at $(a,b)$ if we can find an open ball $B((a,b),\delta)$ in $X \times X$ such that, $(x,y) \in B((a,b),\delta) \implies |d(x,y)-d(a,b)| < \epsilon$. We choose $\delta = \epsilon$.